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PROFESSOR STRANG: This
is my second lecture

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on the example, the model
of graphs and networks.

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Really perfect, beautiful
model for the whole framework.

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And important in itself.

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So, I guess a major thing
that I still have to do

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is discuss A transpose.

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See how A transpose
just naturally appears

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in the balance law.

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Kirchhoff's current
law, KCL, is just

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00:00:55,500 --> 00:00:58,290
like a model for
balance equations.

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00:00:58,290 --> 00:01:02,700
By balance I mean, in
equals out, essentially.

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Flow in equals flow
out because we're

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00:01:05,030 --> 00:01:06,860
talking about steady state.

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So in each node, at
node three, for example,

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I have three edges.

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00:01:14,500 --> 00:01:18,210
So the law, Kirchhoff's
current law at that point

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00:01:18,210 --> 00:01:29,810
is going to tell me that the
flow w_2 plus w_3 minus w_5

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00:01:29,810 --> 00:01:33,150
would be zero if there's
no current source,

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or if I'm feeding current
into there, some current f_3,

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00:01:37,850 --> 00:01:41,000
then it would match f_3.

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00:01:41,000 --> 00:01:43,330
So maybe having just
said those words,

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let me just say, w_2,
I'll say it again,

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w_2 plus w_3 minus
w_5, and I'm hoping

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that I'll find that in
the third column here.

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00:01:56,860 --> 00:01:58,790
And I do.

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00:01:58,790 --> 00:02:02,850
Because I'm thinking,
and we'll write it down,

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00:02:02,850 --> 00:02:04,490
I'm thinking to
take a transpose,

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00:02:04,490 --> 00:02:07,040
so it'll be the
third row, and here I

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00:02:07,040 --> 00:02:10,810
see the w_2, w_3
and minus w_5 that

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00:02:10,810 --> 00:02:15,780
will show up in the
third equation there,

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00:02:15,780 --> 00:02:18,900
the in equals out at node three.

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00:02:18,900 --> 00:02:23,630
Well I'll write that
down, because it's-- So,

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two or three jobs for today.

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00:02:26,400 --> 00:02:32,390
And then Monday I plan to
spend a part of the lecture

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00:02:32,390 --> 00:02:38,620
and all of the review
session in review,

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it's a great chance to go
back to the things we've done

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and collect them, assemble
them, organize them and see them

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00:02:49,860 --> 00:02:50,480
again.

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00:02:50,480 --> 00:02:54,950
So that's my goal for
both sessions on Monday.

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And questions, then,
about every aspect.

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00:02:59,050 --> 00:03:05,600
So, before I get to A
transpose, this part

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00:03:05,600 --> 00:03:09,490
I had written down last time,
but there's a little more

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00:03:09,490 --> 00:03:10,880
to say.

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And what I was saying
and want to continue with

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is the source terms.

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I put in this b but I didn't
draw anything on the network.

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00:03:28,160 --> 00:03:34,010
So let me draw what
these b's represent.

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00:03:34,010 --> 00:03:41,140
So this second, lower, row
is about edge equations,

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edge variables.

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So those batteries
b are on the edges.

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And there will be,
b has length five.

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There are five edges.

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This is a vector of length
five, this matrix, A,

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you remember was five by four.

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It produced from four
inputs, from four potentials

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at the four nodes, A produces
Au, five potential differences.

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So those are the
differences in potentials.

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But then also there
can be source terms

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00:04:18,920 --> 00:04:22,270
from batteries in the
edges, and the minus sign

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is there because I'm
talking about voltage drops.

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So when I say
differences I'm really

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speaking about voltage drops.

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Because that's the
way current flows.

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OK, this is the
moment that I hate.

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00:04:40,560 --> 00:04:44,480
Putting the batteries in.

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00:04:44,480 --> 00:04:49,580
I think you draw a battery
with a long and a short?

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Is that right?

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00:04:50,650 --> 00:04:55,460
And then you put a
plus and a minus?

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Well, can I just say, life
is too short to get those,

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00:05:04,340 --> 00:05:10,930
to get this sign right.

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00:05:10,930 --> 00:05:13,210
You may say when my
car battery stalls

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how do I, because it's
important at that moment, right?

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00:05:15,880 --> 00:05:17,930
When you start it
up you're supposed

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00:05:17,930 --> 00:05:23,330
to put the right battery, the
right lead on the positive

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00:05:23,330 --> 00:05:26,250
and the right on the negative,
or your blow yourself up.

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00:05:26,250 --> 00:05:27,420
OK.

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00:05:27,420 --> 00:05:28,670
So what's my solution?

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Because I literally refuse
to deal with these signs.

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00:05:32,330 --> 00:05:35,220
So my solution is call AAA.

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OK, they're paid to
blow themselves up.

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00:05:37,710 --> 00:05:39,090
They can do it.

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00:05:39,090 --> 00:05:45,360
But so this is serious
now, I don't want

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00:05:45,360 --> 00:05:48,820
to be asked about these signs.

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00:05:48,820 --> 00:05:52,040
And I forgive you for
messing up the signs.

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00:05:52,040 --> 00:05:56,330
So possibly that's plus,
possibly minus, I don't know.

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00:05:56,330 --> 00:06:00,290
But there's a battery
in there of length b_1,

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00:06:00,290 --> 00:06:04,230
of voltage-- A nine-volt
battery, b_1 would be nine.

96
00:06:04,230 --> 00:06:06,780
And depending how it
was placed in there,

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00:06:06,780 --> 00:06:10,510
the b here would be a
plus nine or a minus nine

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00:06:10,510 --> 00:06:12,000
in the first component.

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00:06:12,000 --> 00:06:15,020
And then if I had
another battery here,

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00:06:15,020 --> 00:06:18,990
a b that's on edge four,
there would be a battery b_4,

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00:06:18,990 --> 00:06:24,230
and that would show
up there in Ohm's law.

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00:06:24,230 --> 00:06:27,330
Because Ohm's law will
look at the difference

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00:06:27,330 --> 00:06:30,400
in these potentials,
but then it'll also

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00:06:30,400 --> 00:06:33,790
account for the battery, right?

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00:06:33,790 --> 00:06:39,130
The voltage that comes from
the battery, and somehow

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00:06:39,130 --> 00:06:42,730
combining the Au, which is
the difference in these guys,

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00:06:42,730 --> 00:06:46,760
with the b_4, I'll know
what is the voltage drop

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00:06:46,760 --> 00:06:48,010
across the resistor.

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00:06:48,010 --> 00:06:50,060
And that's what
Ohm's law applies to.

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00:06:50,060 --> 00:06:53,870
Ohm's law says the voltage
drop across the resistor

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00:06:53,870 --> 00:07:00,300
times the conductance, so this
is Ohm's law, that on that edge

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00:07:00,300 --> 00:07:04,220
the voltage drop across the
resistor times the conductance

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00:07:04,220 --> 00:07:05,570
gives the current.

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00:07:05,570 --> 00:07:07,410
So that's the physical law.

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00:07:07,410 --> 00:07:09,470
Is that OK?

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00:07:09,470 --> 00:07:10,850
For batteries.

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00:07:10,850 --> 00:07:12,950
Now a comment on
current sources.

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00:07:12,950 --> 00:07:14,530
So what's with current sources?

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00:07:14,530 --> 00:07:16,180
How would I draw those?

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00:07:16,180 --> 00:07:22,990
Well, very often maybe I
might draw a current going

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00:07:22,990 --> 00:07:28,590
into node one, so that would be
in f_1 that goes into node one

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00:07:28,590 --> 00:07:30,980
and maybe comes out at ground.

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00:07:30,980 --> 00:07:34,010
So that would be a typical f.

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00:07:34,010 --> 00:07:37,850
That if I imposed
a current source,

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00:07:37,850 --> 00:07:40,380
if I sent a current
source through there

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00:07:40,380 --> 00:07:44,490
it would go down and
come out at ground,

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00:07:44,490 --> 00:07:49,480
and our question is what are
the currents in the five edges?

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00:07:49,480 --> 00:07:51,740
What are the potentials
at the four nodes?

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00:07:51,740 --> 00:07:53,890
Well, I'm making
this one ground.

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00:07:53,890 --> 00:07:58,240
So I'm grounding
this one to be u_4=0.

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00:07:58,240 --> 00:08:00,370
And you remember
of course why I had

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00:08:00,370 --> 00:08:04,980
to do that, because this matrix
A transpose A, as it stands

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00:08:04,980 --> 00:08:10,060
is-- What's the matter with
A transpose A as it stands?

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00:08:10,060 --> 00:08:11,960
It's singular, right?

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00:08:11,960 --> 00:08:18,360
And as I, looking ahead,
just a small comment,

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00:08:18,360 --> 00:08:20,640
that this will have
exactly the same thing

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00:08:20,640 --> 00:08:24,030
in so many other applications
in big, finite element codes,

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00:08:24,030 --> 00:08:28,030
you produce a stiffness matrix
that's initially singular.

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00:08:28,030 --> 00:08:30,970
And then you impose the
boundary conditions.

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00:08:30,970 --> 00:08:32,780
That's the efficient
way to do it,

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00:08:32,780 --> 00:08:36,080
is create the matrix
first, that's the big job,

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00:08:36,080 --> 00:08:38,010
then impose the
boundary conditions,

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00:08:38,010 --> 00:08:42,100
that's the small but
occasionally tricky part.

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00:08:42,100 --> 00:08:45,540
Now I indicated
what happened here.

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00:08:45,540 --> 00:08:50,770
When u_4 was zero, that
means that A times u, the u_4

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00:08:50,770 --> 00:08:54,960
there, the zero is multiplying
this, is not unknown any more.

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00:08:54,960 --> 00:08:56,150
It's known.

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00:08:56,150 --> 00:09:03,440
And so that column is not
really any more needed in A.

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00:09:03,440 --> 00:09:11,590
Because u_4 is gone from my
list of unknown potentials.

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00:09:11,590 --> 00:09:18,500
Now, at the same time, when I
went over to A transpose A--

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00:09:18,500 --> 00:09:20,620
I'm making this
comment because there

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00:09:20,620 --> 00:09:22,680
were several good
questions about it.

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00:09:22,680 --> 00:09:30,820
I claim that also,
that row, coming,

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00:09:30,820 --> 00:09:34,030
which of course comes from
the fourth row of A transpose.

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00:09:34,030 --> 00:09:38,640
The fourth column of A is
gone, so the fourth row

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00:09:38,640 --> 00:09:40,800
of A transpose should be gone.

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00:09:40,800 --> 00:09:44,150
And we might just think
why's that, what's going on?

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00:09:44,150 --> 00:09:47,200
Of course, it produces
exactly what we want.

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00:09:47,200 --> 00:09:49,870
It leaves us with a
three by three matrix

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00:09:49,870 --> 00:09:52,000
that's not singular any more.

161
00:09:52,000 --> 00:09:57,140
I've removed that [1, 1, 1, 1]
from the null space by fixing

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00:09:57,140 --> 00:09:58,910
a potential.

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00:09:58,910 --> 00:10:03,850
Grounding a node,
and the problem

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00:10:03,850 --> 00:10:05,990
becomes just what we want.

165
00:10:05,990 --> 00:10:09,350
And I'll write
down the equations

166
00:10:09,350 --> 00:10:15,080
when I get the Kirchhoff current
law to complete the loop.

167
00:10:15,080 --> 00:10:17,880
Now, what's going on?

168
00:10:17,880 --> 00:10:21,910
Let me remove this
current source,

169
00:10:21,910 --> 00:10:28,690
just so we focus on what
I'm speaking about now.

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00:10:28,690 --> 00:10:34,360
On this type of
boundary condition,

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00:10:34,360 --> 00:10:38,590
this is like fixing
a support, right?

172
00:10:38,590 --> 00:10:43,060
It's like fixing a support
in our spring mass problem.

173
00:10:43,060 --> 00:10:46,070
Can I squeeze in a little
spring mass problem,

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00:10:46,070 --> 00:10:56,760
so I have a spring, and then
I'm fixing this displacement,

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00:10:56,760 --> 00:10:58,840
u_4 to be zero.

176
00:10:58,840 --> 00:11:06,220
Now, I want to try
to think through what

177
00:11:06,220 --> 00:11:11,660
happens in a balance
equation, A transpose w,

178
00:11:11,660 --> 00:11:14,850
let me make it A
transpose w equal f,

179
00:11:14,850 --> 00:11:23,410
because that's the case with
right hand side allowed.

180
00:11:23,410 --> 00:11:28,730
And now what happens
when I fix this,

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00:11:28,730 --> 00:11:34,780
I'm asking you to think back
about force balance, which

182
00:11:34,780 --> 00:11:41,370
we certainly saw was an A
transpose w equal f business.

183
00:11:41,370 --> 00:11:47,480
And then parallel will be the
current balance, at that node.

184
00:11:47,480 --> 00:11:54,240
OK, so what was the
deal on force balance?

185
00:11:54,240 --> 00:12:00,480
A transpose with
this fixed in here.

186
00:12:00,480 --> 00:12:04,600
What was the thing with A
transpose that that fixed in?

187
00:12:04,600 --> 00:12:07,880
We did not write the
equation A transpose w equal

188
00:12:07,880 --> 00:12:09,790
f at this point.

189
00:12:09,790 --> 00:12:14,120
We did not write the force
balance equation at that point.

190
00:12:14,120 --> 00:12:19,960
When I fixed u_4, in this case
it was the displacement so I'm

191
00:12:19,960 --> 00:12:24,130
fixing it at zero
displacement, I

192
00:12:24,130 --> 00:12:27,390
didn't have a force balance
in the A transpose part

193
00:12:27,390 --> 00:12:28,680
at this node.

194
00:12:28,680 --> 00:12:30,670
Now, you could say why?

195
00:12:30,670 --> 00:12:33,310
Because of course
forces have to balance.

196
00:12:33,310 --> 00:12:35,020
But what's going on?

197
00:12:35,020 --> 00:12:39,430
What's really happening here
is I don't have to write out,

198
00:12:39,430 --> 00:12:43,680
I don't have to-- the
displacement here is known.

199
00:12:43,680 --> 00:12:45,700
It's not unknown.

200
00:12:45,700 --> 00:12:50,260
And let me say it in a sentence.

201
00:12:50,260 --> 00:12:54,530
Force balance does hold
because the support

202
00:12:54,530 --> 00:13:01,890
supplies whatever force it takes
to balance the internal forces.

203
00:13:01,890 --> 00:13:09,120
So, in other words, let me say
that again, the force balance

204
00:13:09,120 --> 00:13:11,720
will hold and it will
tell me, after I've

205
00:13:11,720 --> 00:13:14,120
solved the rest of the
problem, it'll tell me

206
00:13:14,120 --> 00:13:18,850
what the support has to supply,
how much force the support is

207
00:13:18,850 --> 00:13:19,720
actually supplying.

208
00:13:19,720 --> 00:13:22,430
It's a reaction force,
it would be called.

209
00:13:22,430 --> 00:13:26,710
So a reaction force is
whatever the support has

210
00:13:26,710 --> 00:13:29,530
to do to fix that displacement.

211
00:13:29,530 --> 00:13:35,810
And so I solved
the fixed problems

212
00:13:35,810 --> 00:13:40,020
for all the other displacements
and all the spring forces.

213
00:13:40,020 --> 00:13:42,320
And then I could
come back at the end

214
00:13:42,320 --> 00:13:46,060
and say OK, what was the force
in that spring and therefore

215
00:13:46,060 --> 00:13:51,340
how much is that support,
what's the force being supplied

216
00:13:51,340 --> 00:13:52,770
by that support.

217
00:13:52,770 --> 00:13:53,920
See what I'm saying?

218
00:13:53,920 --> 00:14:05,270
That the force balance at
this node comes afterwards.

219
00:14:05,270 --> 00:14:12,020
That equation is like knocking
that one out of this problem.

220
00:14:12,020 --> 00:14:17,860
It would be the same here,
I fix the potential at zero.

221
00:14:17,860 --> 00:14:19,760
I fix ground at zero.

222
00:14:19,760 --> 00:14:21,960
Current flows.

223
00:14:21,960 --> 00:14:26,540
Maybe some, maybe I might
fix that potential at one.

224
00:14:26,540 --> 00:14:28,350
I fix that potential at zero.

225
00:14:28,350 --> 00:14:29,770
Current flows.

226
00:14:29,770 --> 00:14:31,010
OK.

227
00:14:31,010 --> 00:14:33,730
Ta-da, I find out, I
compute what they are,

228
00:14:33,730 --> 00:14:40,200
and this row and
column will be gone.

229
00:14:40,200 --> 00:14:41,620
Find out what they are.

230
00:14:41,620 --> 00:14:46,840
At this node, what's happened?

231
00:14:46,840 --> 00:14:50,010
What's happening to the
balance of currents?

232
00:14:50,010 --> 00:14:52,820
Does this current have
to balance that current

233
00:14:52,820 --> 00:14:54,220
at the ground?

234
00:14:54,220 --> 00:14:55,070
No.

235
00:14:55,070 --> 00:15:00,700
The ground, whatever
current comes here

236
00:15:00,700 --> 00:15:04,110
plus whatever current comes
here, goes into ground.

237
00:15:04,110 --> 00:15:07,920
Do you see the point?

238
00:15:07,920 --> 00:15:10,200
Kirchhoff's current
law, the balance

239
00:15:10,200 --> 00:15:14,840
of currents that in
equals out, is true

240
00:15:14,840 --> 00:15:18,000
but it's not an
equation I have to solve

241
00:15:18,000 --> 00:15:20,920
in finding the
currents, it's something

242
00:15:20,920 --> 00:15:22,540
I can discover at the end.

243
00:15:22,540 --> 00:15:25,860
I can say OK, how much
current flowed from ground?

244
00:15:25,860 --> 00:15:27,280
And similarly up here.

245
00:15:27,280 --> 00:15:32,167
If I fix u_1 to be
zero, then you, no, I

246
00:15:32,167 --> 00:15:34,500
don't want to fix it to be
zero too, that would be a way

247
00:15:34,500 --> 00:15:36,850
to make very little happen.

248
00:15:36,850 --> 00:15:41,940
Let me fix u_1 to
be one, so this is

249
00:15:41,940 --> 00:15:44,810
a standard important problem.

250
00:15:44,810 --> 00:15:48,210
It's like what's the
resistance in the net-- what's

251
00:15:48,210 --> 00:15:50,870
the net system resistance?

252
00:15:50,870 --> 00:15:55,500
If I fix this at one fix this
at zero, some current will flow,

253
00:15:55,500 --> 00:15:59,740
it'll come out here.

254
00:15:59,740 --> 00:16:01,800
And that'll be the
current going into here.

255
00:16:01,800 --> 00:16:06,680
Somehow there'll be a balance
there, and a balance there.

256
00:16:06,680 --> 00:16:10,680
But it's found later.

257
00:16:10,680 --> 00:16:14,220
Just the way the force in
the support is found later.

258
00:16:14,220 --> 00:16:16,430
OK, that takes a
little thinking.

259
00:16:16,430 --> 00:16:23,400
I just wanted to, because I had
blithely knocked this row out.

260
00:16:23,400 --> 00:16:26,000
And you could do
it on the basis,

261
00:16:26,000 --> 00:16:28,740
well if you knock
out this column of A

262
00:16:28,740 --> 00:16:31,360
then you're knocking out
the row of A transpose,

263
00:16:31,360 --> 00:16:35,050
and therefore A transpose
A will be three by three.

264
00:16:35,050 --> 00:16:40,340
And it'll go down to two by
two if I fix that potential.

265
00:16:40,340 --> 00:16:43,730
And if I don't fix it at
zero, if I fix it at one

266
00:16:43,730 --> 00:16:47,180
then something will move
to the right hand side.

267
00:16:47,180 --> 00:16:57,590
OK. that's that point I
hope at least discussed.

268
00:16:57,590 --> 00:17:03,220
So now we have the
whole pattern here.

269
00:17:03,220 --> 00:17:09,480
Except that I really have still
to justify the fact that it

270
00:17:09,480 --> 00:17:14,650
truly is A transpose, the
transpose of that matrix,

271
00:17:14,650 --> 00:17:17,010
that comes into
Kirchhoff's current law.

272
00:17:17,010 --> 00:17:20,200
I guess in the first
minute of the lecture I

273
00:17:20,200 --> 00:17:21,800
looked at that particular node.

274
00:17:21,800 --> 00:17:23,410
Let's look at all the nodes.

275
00:17:23,410 --> 00:17:28,100
Let me look at A
transpose w equals zero.

276
00:17:28,100 --> 00:17:28,960
OK.

277
00:17:28,960 --> 00:17:32,150
So I'll copy a transpose
because I truly

278
00:17:32,150 --> 00:17:35,790
believe that Kirchhoff
would want me to do it.

279
00:17:35,790 --> 00:17:45,080
OK, so that becomes a row, that
becomes the next row, of course

280
00:17:45,080 --> 00:17:50,500
I see three guys
going into node one,

281
00:17:50,500 --> 00:17:53,370
then the one that
I looked at before.

282
00:17:53,370 --> 00:17:59,310
That's the three edges, node
three, and then the last one.

283
00:17:59,310 --> 00:18:02,340
Let's keep the last
one in for now.

284
00:18:02,340 --> 00:18:08,310
And because if I didn't fix
that I'd have that last one.

285
00:18:08,310 --> 00:18:12,290
There we go, so that multiplies,
what does that multiply now

286
00:18:12,290 --> 00:18:14,020
in Kirchhoff's current law?

287
00:18:14,020 --> 00:18:16,680
Multiplies w, so currents.

288
00:18:16,680 --> 00:18:22,480
So here the currents, one,
two, three, four and five.

289
00:18:22,480 --> 00:18:26,300
All I'm doing now is
just, like, convincing you

290
00:18:26,300 --> 00:18:29,230
that it really is a
transpose, that if I look at,

291
00:18:29,230 --> 00:18:32,250
let me pick that
node now, if I look

292
00:18:32,250 --> 00:18:38,840
at that node I see
edge one coming in,

293
00:18:38,840 --> 00:18:42,820
I see edge three going out,
I see edge four going out

294
00:18:42,820 --> 00:18:45,410
and when I look at
that second node,

295
00:18:45,410 --> 00:18:48,980
I look here at the second
row, I see edge one coming in,

296
00:18:48,980 --> 00:18:51,020
edge three and edge
four going out,

297
00:18:51,020 --> 00:18:54,280
multiplying those currents,
and that will give,

298
00:18:54,280 --> 00:18:58,580
that current balance
there gives that zero.

299
00:18:58,580 --> 00:18:59,080
Right?

300
00:18:59,080 --> 00:19:02,510
That current balance
at node two gives that

301
00:19:02,510 --> 00:19:05,770
zero in the right hand side.

302
00:19:05,770 --> 00:19:09,720
And then, of course,
other zeroes are here too.

303
00:19:09,720 --> 00:19:14,860
This is minus w_1, minus
w_2, so I have a zero.

304
00:19:14,860 --> 00:19:19,240
This is with no current sources.

305
00:19:19,240 --> 00:19:19,930
OK.

306
00:19:19,930 --> 00:19:23,890
And this one, of course.

307
00:19:23,890 --> 00:19:29,070
I guess I'm hoping that you
say yes, the A transpose really

308
00:19:29,070 --> 00:19:35,010
was the right matrix to express
in equals out, Kirchhoff's

309
00:19:35,010 --> 00:19:38,040
current law.

310
00:19:38,040 --> 00:19:41,020
OK.

311
00:19:41,020 --> 00:19:43,620
Of course, by now you're
probably getting blasé.

312
00:19:43,620 --> 00:19:47,600
You expect it to be A transpose,
you don't need any convincing.

313
00:19:47,600 --> 00:19:50,980
But it's kind of good to
see each time because it's

314
00:19:50,980 --> 00:19:53,620
like, well, it's not a miracle.

315
00:19:53,620 --> 00:19:56,100
But it's like a miracle.

316
00:19:56,100 --> 00:19:57,930
It's as good as a miracle.

317
00:19:57,930 --> 00:20:01,930
Because to get A
transpose is just

318
00:20:01,930 --> 00:20:04,590
what makes everything right.

319
00:20:04,590 --> 00:20:11,340
Now, here's a question.

320
00:20:11,340 --> 00:20:15,830
This is a question
worth thinking about.

321
00:20:15,830 --> 00:20:17,120
What are the solutions?

322
00:20:17,120 --> 00:20:21,540
If I only look at this
piece of the framework,

323
00:20:21,540 --> 00:20:25,330
if I just look at Kirchhoff's
current law, it's telling me,

324
00:20:25,330 --> 00:20:27,950
and I have zero current sources.

325
00:20:27,950 --> 00:20:30,410
Well let's take
that, let me take

326
00:20:30,410 --> 00:20:35,210
just this piece of the
whole framework and ask you,

327
00:20:35,210 --> 00:20:41,210
how many vectors w,
how many solutions?

328
00:20:41,210 --> 00:20:42,780
Are there any solutions?

329
00:20:42,780 --> 00:20:45,510
Well, of course, there's
always the zero solution.

330
00:20:45,510 --> 00:20:50,300
But I always ask you, what
are the solutions when

331
00:20:50,300 --> 00:20:52,790
zero is on the right hand side?

332
00:20:52,790 --> 00:20:56,770
So, what are the w's that
solve Kirchhoff's current law.

333
00:20:56,770 --> 00:20:58,390
Now that's a new question.

334
00:20:58,390 --> 00:20:59,930
We haven't asked that before.

335
00:20:59,930 --> 00:21:04,880
What we asked before was
the question Au, Au=0,

336
00:21:04,880 --> 00:21:09,590
remember it was then A.
This five by four matrix,

337
00:21:09,590 --> 00:21:11,880
u had four components.

338
00:21:11,880 --> 00:21:16,720
Just remind me, and I'm
putting that column back in,

339
00:21:16,720 --> 00:21:19,840
so this is still here.

340
00:21:19,840 --> 00:21:23,760
In the un-reduced,
un-grounded case.

341
00:21:23,760 --> 00:21:27,090
Well, just so we get
started, remind me

342
00:21:27,090 --> 00:21:35,120
what the solutions u were for
Au equals zero. [1, 1, 1, 1].

343
00:21:35,120 --> 00:21:36,330
Or any multiple of it.

344
00:21:36,330 --> 00:21:39,280
A whole line of vectors.

345
00:21:39,280 --> 00:21:42,090
[c, c, c, c], any constant c.

346
00:21:42,090 --> 00:21:47,040
OK, so this had, there
were four columns.

347
00:21:47,040 --> 00:21:49,140
But only three were independent.

348
00:21:49,140 --> 00:21:49,900
OK.

349
00:21:49,900 --> 00:21:56,580
Now, now I've made
them into rows.

350
00:21:56,580 --> 00:21:59,170
And I made the
rows into columns.

351
00:21:59,170 --> 00:22:02,640
So now I have five columns.

352
00:22:02,640 --> 00:22:08,240
What I'm leading to is, I
want to count in advance

353
00:22:08,240 --> 00:22:12,900
how many w's I should be looking
for, and then look for them.

354
00:22:12,900 --> 00:22:18,420
OK, so the first question is
how many different solutions w.

355
00:22:18,420 --> 00:22:22,280
First of all, are
there some solutions?

356
00:22:22,280 --> 00:22:25,880
Is there is a solution w,
other than zero of course,

357
00:22:25,880 --> 00:22:27,980
to this system?

358
00:22:27,980 --> 00:22:30,950
Well, as we said
last time, we've

359
00:22:30,950 --> 00:22:33,180
only got four equations here.

360
00:22:33,180 --> 00:22:35,810
We've got five unknowns.

361
00:22:35,810 --> 00:22:38,490
Of course there's at
least one solution.

362
00:22:38,490 --> 00:22:42,190
Four equations, five unknowns,
I can do elimination,

363
00:22:42,190 --> 00:22:45,490
whatever systematic
procedure you want me to do.

364
00:22:45,490 --> 00:22:48,030
In the end, I'm going
to find a solution.

365
00:22:48,030 --> 00:22:50,100
Now, I might find
more solutions.

366
00:22:50,100 --> 00:22:52,590
So that's the question.

367
00:22:52,590 --> 00:22:57,840
So what's the, now this is the
key fact of linear algebra.

368
00:22:57,840 --> 00:23:02,930
Which is, it just tells us
the numbers of everything.

369
00:23:02,930 --> 00:23:07,950
So this told us that there
were three independent columns.

370
00:23:07,950 --> 00:23:15,130
Of A. Now, for that key
theorem, which tells me that,

371
00:23:15,130 --> 00:23:19,040
how many independent
rows of A are there?

372
00:23:19,040 --> 00:23:19,970
Three.

373
00:23:19,970 --> 00:23:22,940
That number is equal.

374
00:23:22,940 --> 00:23:25,860
I just want to say that's
a pretty remarkable fact.

375
00:23:25,860 --> 00:23:32,930
If I have a 50 by 80 matrix
and that 50 by 80 matrix

376
00:23:32,930 --> 00:23:38,240
has 17 independent columns,
then this great fact

377
00:23:38,240 --> 00:23:41,330
tells me that there are
17 independent rows.

378
00:23:41,330 --> 00:23:47,960
And if those 50 times 80,
4,000 numbers are random,

379
00:23:47,960 --> 00:23:52,690
boy, you can't look at
it and see what are they.

380
00:23:52,690 --> 00:23:53,740
Independent rows.

381
00:23:53,740 --> 00:23:56,660
But if you know there are
17 independent columns then

382
00:23:56,660 --> 00:23:58,570
there are 17 independent rows.

383
00:23:58,570 --> 00:24:01,990
So, what does that mean here?

384
00:24:01,990 --> 00:24:04,450
That means that out
of the five columns

385
00:24:04,450 --> 00:24:10,810
of A transpose, which were rows
of A, three are independent.

386
00:24:10,810 --> 00:24:14,220
So just tell me, how many
solutions I'm looking for.

387
00:24:14,220 --> 00:24:16,470
Before I look for them.

388
00:24:16,470 --> 00:24:20,750
So the number of-- The
number of independent w's,

389
00:24:20,750 --> 00:24:25,700
independent solutions
will be what?

390
00:24:25,700 --> 00:24:27,770
What's your guess?

391
00:24:27,770 --> 00:24:28,620
Two!

392
00:24:28,620 --> 00:24:30,320
Two is the right guess.

393
00:24:30,320 --> 00:24:34,700
Two, because I have
altogether five unknowns,

394
00:24:34,700 --> 00:24:39,650
I subtract three equations
that are really there,

395
00:24:39,650 --> 00:24:41,670
I have three real
equations there

396
00:24:41,670 --> 00:24:43,300
even though it looks like four.

397
00:24:43,300 --> 00:24:44,940
And I get two.

398
00:24:44,940 --> 00:24:49,460
So that general
picture is n-- Oh, I'm

399
00:24:49,460 --> 00:24:54,070
sorry it's actually m because
I'm doing the transpose here.

400
00:24:54,070 --> 00:25:00,890
So it's m w's minus r, the rank.

401
00:25:00,890 --> 00:25:03,840
So that's m is five,
the rank is three

402
00:25:03,840 --> 00:25:07,520
and this counts the number
of independent solutions.

403
00:25:07,520 --> 00:25:12,960
So it's a nice, it
couldn't be better.

404
00:25:12,960 --> 00:25:17,520
It's a fundamental count of
how many solutions are there.

405
00:25:17,520 --> 00:25:22,090
You're really taking the
number of unknowns, five,

406
00:25:22,090 --> 00:25:24,490
and you're subtracting
the number of equations

407
00:25:24,490 --> 00:25:27,270
that are really there,
three, and that leaves you

408
00:25:27,270 --> 00:25:30,780
with two solutions which we
will have to find in a minute.

409
00:25:30,780 --> 00:25:35,930
Can you say why there's really
only three equations there?

410
00:25:35,930 --> 00:25:41,910
Why do I say that that
fourth equation is not

411
00:25:41,910 --> 00:25:45,600
contributing anything new?

412
00:25:45,600 --> 00:25:50,040
I believe that that fourth
equation is a consequence

413
00:25:50,040 --> 00:25:52,400
of the first three.

414
00:25:52,400 --> 00:25:55,830
And, therefore, if it's
there or if it's not there,

415
00:25:55,830 --> 00:25:59,490
it's not telling me
anything new about in=out.

416
00:25:59,490 --> 00:26:03,060
In other words, if I
have a closed system,

417
00:26:03,060 --> 00:26:05,700
closed because it's
zero on the right side,

418
00:26:05,700 --> 00:26:10,430
if I have a closed system and
I have in=out at those three

419
00:26:10,430 --> 00:26:14,400
nodes, then I'll automatically
have in=out at the fourth node,

420
00:26:14,400 --> 00:26:19,060
because the total in is zero.

421
00:26:19,060 --> 00:26:20,690
And the total out is zero.

422
00:26:20,690 --> 00:26:23,850
So if I'm right at three I'll
be right at the fourth one.

423
00:26:23,850 --> 00:26:26,360
And now just tell
me with the numbers,

424
00:26:26,360 --> 00:26:37,050
how would I get this equation
w_4+w_5=0 from the first three?

425
00:26:37,050 --> 00:26:39,450
Well, it's probably
the same way that I

426
00:26:39,450 --> 00:26:41,400
got that column,
that that column was

427
00:26:41,400 --> 00:26:44,920
connected to those columns.

428
00:26:44,920 --> 00:26:46,940
What do I do?

429
00:26:46,940 --> 00:26:48,900
Add them.

430
00:26:48,900 --> 00:26:52,640
If you add that equation to
that equation to that equation,

431
00:26:52,640 --> 00:26:54,930
add those three
equations, what happens?

432
00:26:54,930 --> 00:26:59,320
the w_1's cancel, the w_2's
cancel, the w_3's cancel,

433
00:26:59,320 --> 00:27:02,940
this says there's a
minus w_4 and a minus w_5

434
00:27:02,940 --> 00:27:07,280
that adds to zero plus
zero plus zero, minus w_4,

435
00:27:07,280 --> 00:27:11,850
minus w_5 equalling zero
is the same as plus w_4

436
00:27:11,850 --> 00:27:14,930
plus w_5 equals zero.

437
00:27:14,930 --> 00:27:18,380
The four equations add
to zero equals zero.

438
00:27:18,380 --> 00:27:21,530
That's what the
central thing is.

439
00:27:21,530 --> 00:27:24,210
The four equations add
to zero equals zero just

440
00:27:24,210 --> 00:27:28,740
the way the four columns up
here added to the zero column.

441
00:27:28,740 --> 00:27:29,730
OK.

442
00:27:29,730 --> 00:27:32,070
So we got the count.

443
00:27:32,070 --> 00:27:37,010
Now, this is the
interesting part, always.

444
00:27:37,010 --> 00:27:38,860
What are the solutions?

445
00:27:38,860 --> 00:27:40,900
What are the actual w's?

446
00:27:40,900 --> 00:27:43,570
OK.

447
00:27:43,570 --> 00:27:45,890
You could say, wait
a minute, you're

448
00:27:45,890 --> 00:27:52,210
asking me to solve four
equations and five unknowns,

449
00:27:52,210 --> 00:27:53,710
and just say what the answer is.

450
00:27:53,710 --> 00:27:57,010
Well, normally that's
not reasonable.

451
00:27:57,010 --> 00:28:02,140
But here we can get
help from the graph.

452
00:28:02,140 --> 00:28:04,570
Let me give you an idea here.

453
00:28:04,570 --> 00:28:09,380
So what are we looking for
on the graph, that solves it?

454
00:28:09,380 --> 00:28:11,760
We're looking for
a bunch of currents

455
00:28:11,760 --> 00:28:14,860
that balance themselves.

456
00:28:14,860 --> 00:28:15,360
Right?

457
00:28:15,360 --> 00:28:17,490
We've got zero on
the right hand side.

458
00:28:17,490 --> 00:28:20,810
So we're not getting
any help from outside.

459
00:28:20,810 --> 00:28:24,200
How could you send
current in this loop

460
00:28:24,200 --> 00:28:27,750
in a way that would
satisfy Kirchhoff.

461
00:28:27,750 --> 00:28:29,650
He'd be happy.

462
00:28:29,650 --> 00:28:32,270
The balance law would be true.

463
00:28:32,270 --> 00:28:37,810
OK. b-- I'm just
looking at currents.

464
00:28:37,810 --> 00:28:41,380
Current w_1, w_2, w_3.

465
00:28:41,380 --> 00:28:44,010
Is there any
combination of w_1, w_2,

466
00:28:44,010 --> 00:28:52,510
w_3 that would balance itself,
that would make Kirchhoff OK?

467
00:28:52,510 --> 00:28:55,820
Well, here's the idea.

468
00:28:55,820 --> 00:28:58,620
Send that current around a loop.

469
00:28:58,620 --> 00:29:05,120
Loop currents are solutions
to Kirchhoff's balance law.

470
00:29:05,120 --> 00:29:09,970
If I send an amp on
that edge, on that edge

471
00:29:09,970 --> 00:29:13,640
and backward on
that edge, right?

472
00:29:13,640 --> 00:29:17,830
It's around a loop at every
node, it's totally OK.

473
00:29:17,830 --> 00:29:21,310
So I believe that a
particular solution will

474
00:29:21,310 --> 00:29:29,280
be for these things to be,
let's see what did I say? w_1,

475
00:29:29,280 --> 00:29:34,710
I'll send one around. w_3 will
be a one. w_2 was backwards,

476
00:29:34,710 --> 00:29:37,260
it wasn't traveling on w_4.

477
00:29:37,260 --> 00:29:39,790
I think that's a solution.

478
00:29:39,790 --> 00:29:42,180
That loop current
gives me a solution so

479
00:29:42,180 --> 00:29:49,340
let me call this solution,
that's the first solution.

480
00:29:49,340 --> 00:29:51,910
And if we do these
multiplications, of course

481
00:29:51,910 --> 00:29:53,670
it's going to come out right.

482
00:29:53,670 --> 00:29:58,930
OK, so that's
solution number one.

483
00:29:58,930 --> 00:30:01,270
A w that works.

484
00:30:01,270 --> 00:30:04,490
OK, with that hint, tell
me a second w that works.

485
00:30:04,490 --> 00:30:08,030
In fact, since
there are only two,

486
00:30:08,030 --> 00:30:12,820
you'll be giving me
the rest of the answer.

487
00:30:12,820 --> 00:30:17,680
So that was a loop current
that went around that loop.

488
00:30:17,680 --> 00:30:20,730
Tell me another one.

489
00:30:20,730 --> 00:30:25,640
Well, we're a big class
but everybody's seen it.

490
00:30:25,640 --> 00:30:30,620
How about around that loop?

491
00:30:30,620 --> 00:30:33,540
So that would be
another thing and it's

492
00:30:33,540 --> 00:30:37,620
pretty clearly not
the same as this one.

493
00:30:37,620 --> 00:30:41,390
So I'm really truly finding
two independent solutions.

494
00:30:41,390 --> 00:30:43,670
And what is that
second solution?

495
00:30:43,670 --> 00:30:48,810
Let me maybe just
put it in here.

496
00:30:48,810 --> 00:30:52,860
And they're both giving
me [0,  0, 0, 0].

497
00:30:52,860 --> 00:30:57,560
And now, what is this number two
solution, the loop number two?

498
00:30:57,560 --> 00:31:00,790
One and two are
not involved now.

499
00:31:00,790 --> 00:31:07,900
Number three, see I'm usually
sending it counterclockwise,

500
00:31:07,900 --> 00:31:09,430
that's the sort of convention.

501
00:31:09,430 --> 00:31:13,460
But you know, of
course you just have

502
00:31:13,460 --> 00:31:17,270
to follow some convention in
connection with the arrows.

503
00:31:17,270 --> 00:31:20,570
So that would go backwards
on three, I think.

504
00:31:20,570 --> 00:31:24,630
Forwards on four, and
backwards on five.

505
00:31:24,630 --> 00:31:27,770
So that would be
solution number two.

506
00:31:27,770 --> 00:31:29,650
OK.

507
00:31:29,650 --> 00:31:32,040
Now, tell me what all
the solutions are.

508
00:31:32,040 --> 00:31:36,960
I found two particular
solutions, two particular loop

509
00:31:36,960 --> 00:31:39,970
currents, particularly easy.

510
00:31:39,970 --> 00:31:45,090
What would be all the solutions
to Kirchhoff's current law,

511
00:31:45,090 --> 00:31:47,800
A transpose w equals zero?

512
00:31:47,800 --> 00:31:50,980
Every w now since I've
found the right number,

513
00:31:50,980 --> 00:31:57,840
every w will be a
combination of those two.

514
00:31:57,840 --> 00:31:59,900
Ah, well wait a minute.

515
00:31:59,900 --> 00:32:01,110
Have I got them all?

516
00:32:01,110 --> 00:32:04,720
I should have thought,
what about current

517
00:32:04,720 --> 00:32:07,660
around the big loop?

518
00:32:07,660 --> 00:32:10,870
That would certainly
satisfy Kirchhoff.

519
00:32:10,870 --> 00:32:19,560
Plus one, one, so why is
this not number three?

520
00:32:19,560 --> 00:32:22,690
Around the big loop
I have a one and then

521
00:32:22,690 --> 00:32:24,760
a one on the fourth position.

522
00:32:24,760 --> 00:32:28,420
Backwards on five,
backwards on two.

523
00:32:28,420 --> 00:32:32,310
And so there is number--
So I'll put number three

524
00:32:32,310 --> 00:32:36,130
with a question mark.

525
00:32:36,130 --> 00:32:41,670
What's up with that guy?

526
00:32:41,670 --> 00:32:46,510
You know, unless mathematics
has got to close up shop,

527
00:32:46,510 --> 00:32:51,070
this better be a
combination of those.

528
00:32:51,070 --> 00:32:52,170
And of course, it is.

529
00:32:52,170 --> 00:32:56,090
If I send something around
the top loop and something

530
00:32:56,090 --> 00:32:58,440
around the second loop
and add them together,

531
00:32:58,440 --> 00:33:00,740
they'll cancel on
the middle edge

532
00:33:00,740 --> 00:33:04,000
there and produce number three.

533
00:33:04,000 --> 00:33:07,580
So this is probably
just the sum.

534
00:33:07,580 --> 00:33:11,240
If I add that one to that one,
I think I get number three.

535
00:33:11,240 --> 00:33:14,270
So number three is true.

536
00:33:14,270 --> 00:33:17,510
It's a solution but
it's not a new one.

537
00:33:17,510 --> 00:33:20,590
OK.

538
00:33:20,590 --> 00:33:25,720
That was simple, right, once we
saw that loops gave the answer.

539
00:33:25,720 --> 00:33:33,330
But, it's, actually
it's quite important

540
00:33:33,330 --> 00:33:35,570
and appears everywhere.

541
00:33:35,570 --> 00:33:42,400
In fact the theory of
electrical networks,

542
00:33:42,400 --> 00:33:43,880
current laws and so on.

543
00:33:43,880 --> 00:33:46,060
I mean that used to be
a, like, basic course

544
00:33:46,060 --> 00:33:47,340
in electrical engineering.

545
00:33:47,340 --> 00:33:52,220
There was a text by Professor
Ernst Guillemin, I remember.

546
00:33:52,220 --> 00:33:56,420
It's sort of not so central
to the world any more.

547
00:33:56,420 --> 00:34:03,260
And now-- Bu the structure
is just right somehow.

548
00:34:03,260 --> 00:34:08,040
And what I wanted to say is
you could take, in those days

549
00:34:08,040 --> 00:34:11,190
you maybe took loop
currents as the unknowns.

550
00:34:11,190 --> 00:34:13,510
You could think of
currents in the loops

551
00:34:13,510 --> 00:34:16,300
as your principal unknowns.

552
00:34:16,300 --> 00:34:17,620
We don't do that now.

553
00:34:17,620 --> 00:34:23,400
But, oh, there's, yeah
it comes up again.

554
00:34:23,400 --> 00:34:27,250
Knowing all the solutions to
A transpose w equals zero,

555
00:34:27,250 --> 00:34:31,350
well, you'll see, what's ahead?

556
00:34:31,350 --> 00:34:36,430
It will be the continuous analog
of this, where I have flows,

557
00:34:36,430 --> 00:34:40,970
not just around a
graph, but in a region.

558
00:34:40,970 --> 00:34:43,510
And Laplace's equation
is going to come up,

559
00:34:43,510 --> 00:34:46,830
and the equations of
divergence and gradient,

560
00:34:46,830 --> 00:34:50,830
all that great stuff
is coming in Chapter 3.

561
00:34:50,830 --> 00:34:59,160
And this is somehow
the discrete case.

562
00:34:59,160 --> 00:35:03,000
These loop currents, that has
something to do with the curl.

563
00:35:03,000 --> 00:35:07,560
And these differences
that A takes

564
00:35:07,560 --> 00:35:09,800
has something to
do with gradients.

565
00:35:09,800 --> 00:35:12,890
And this, Kirchhoff's
current law

566
00:35:12,890 --> 00:35:15,580
has something to
do with divergence.

567
00:35:15,580 --> 00:35:21,010
Can I just say ahead of time,
what we're doing is really

568
00:35:21,010 --> 00:35:27,270
good to see and get it
because then you have a way

569
00:35:27,270 --> 00:35:33,390
to understand vector calculus.

570
00:35:33,390 --> 00:35:36,410
This is discrete vector
calculus we're doing.

571
00:35:36,410 --> 00:35:37,170
OK.

572
00:35:37,170 --> 00:35:40,810
Now, it's just right.

573
00:35:40,810 --> 00:35:45,290
Forgive me for my sermon here.

574
00:35:45,290 --> 00:35:45,910
Alright.

575
00:35:45,910 --> 00:35:50,980
Now, may I bring the
pieces together finally?

576
00:35:50,980 --> 00:35:56,930
May I bring the three
steps together into,

577
00:35:56,930 --> 00:36:01,910
well first you would say
bring them into one equation.

578
00:36:01,910 --> 00:36:05,880
Put the three steps, combine
the three steps into one.

579
00:36:05,880 --> 00:36:06,800
OK.

580
00:36:06,800 --> 00:36:10,760
So, what happens if I do that?

581
00:36:10,760 --> 00:36:18,250
I take that last step A
transpose-- f is A transpose w.

582
00:36:18,250 --> 00:36:24,880
So now I'm going to
get one equation.

583
00:36:24,880 --> 00:36:26,790
Which is the
equation that's going

584
00:36:26,790 --> 00:36:30,320
to have the stiffness matrix
in it, A transpose C A.

585
00:36:30,320 --> 00:36:35,380
It's the conductance matrix.

586
00:36:35,380 --> 00:36:38,050
And it's the equation
that a big finite element

587
00:36:38,050 --> 00:36:43,490
code, a circuit simulation
code-- It's the matrix

588
00:36:43,490 --> 00:36:45,870
they have to find and work with.

589
00:36:45,870 --> 00:36:50,450
And those codes are enormous,
and SPICE by the way

590
00:36:50,450 --> 00:36:54,540
is the sort of grandfather
of circuit simulation codes.

591
00:36:54,540 --> 00:36:57,750
Somebody at Berkeley had
the sense to see hey,

592
00:36:57,750 --> 00:37:00,700
we've got giant circuits now.

593
00:37:00,700 --> 00:37:03,730
Modern circuits have
thousands of elements.

594
00:37:03,730 --> 00:37:09,160
And you can't do it by eye the
way we can do this one by eye.

595
00:37:09,160 --> 00:37:12,780
You've got to organize
it and write a code,

596
00:37:12,780 --> 00:37:14,660
and SPICE is the start.

597
00:37:14,660 --> 00:37:18,530
So one way to do it is to
end up with one equation.

598
00:37:18,530 --> 00:37:22,210
So that was f equals
A transpose w.

599
00:37:22,210 --> 00:37:25,550
I'm now going, I'm just
assembling the whole loop.

600
00:37:25,550 --> 00:37:28,500
But w is Ce.

601
00:37:28,500 --> 00:37:41,400
So that's A transpose
Ce, but e is b minus Au.

602
00:37:41,400 --> 00:37:43,750
Nothing new there.

603
00:37:43,750 --> 00:37:49,290
Nothing new maybe except that it
involves both the f and the b,

604
00:37:49,290 --> 00:37:52,180
where our earlier
examples involved either

605
00:37:52,180 --> 00:37:56,920
an f, in masses and springs,
or a b in the squares.

606
00:37:56,920 --> 00:37:58,060
Now they're both here.

607
00:37:58,060 --> 00:38:01,760
So now, that's my
equation, f equals this.

608
00:38:01,760 --> 00:38:06,600
And now let me just move that to
the, let me just recollect it,

609
00:38:06,600 --> 00:38:10,180
that's A transpose C A, the
big thing that I wanted to see.

610
00:38:10,180 --> 00:38:13,220
I'll put it on the left side.

611
00:38:13,220 --> 00:38:15,560
And what will I have
on the right side?

612
00:38:15,560 --> 00:38:21,740
I'll have A transpose C b.

613
00:38:21,740 --> 00:38:25,050
And I'll have, f is now
coming over to the other side

614
00:38:25,050 --> 00:38:26,710
with a minus.

615
00:38:26,710 --> 00:38:28,360
Minus f.

616
00:38:28,360 --> 00:38:31,840
That's the big equation.

617
00:38:31,840 --> 00:38:33,840
You could say that's
the fundamental equation

618
00:38:33,840 --> 00:38:35,440
of equilibrium.

619
00:38:35,440 --> 00:38:38,030
And you see how it's right?

620
00:38:38,030 --> 00:38:43,230
It involves the A transpose
C A, which we expect.

621
00:38:43,230 --> 00:38:45,960
Over here was the
A transpose C b.

622
00:38:45,960 --> 00:38:51,760
Now, which problem,
before networks,

623
00:38:51,760 --> 00:38:56,130
produced an A transpose
b or an A transpose C b?

624
00:38:56,130 --> 00:38:57,750
That was least squares.

625
00:38:57,750 --> 00:39:01,540
And now, so that's the
least squares, the b term.

626
00:39:01,540 --> 00:39:03,570
The b is there with
a couple of matrices

627
00:39:03,570 --> 00:39:07,660
because b entered the
problem just one step around.

628
00:39:07,660 --> 00:39:09,930
It had two more steps to go.

629
00:39:09,930 --> 00:39:13,160
It had a C step and then
an A transpose step.

630
00:39:13,160 --> 00:39:16,040
Whereas f up here is
at the very end and now

631
00:39:16,040 --> 00:39:19,310
it appears with a minus sign.

632
00:39:19,310 --> 00:39:22,810
That's different from
springs and masses

633
00:39:22,810 --> 00:39:26,510
simply because the sign
conventions, you could say.

634
00:39:26,510 --> 00:39:28,480
OK, there is the equation.

635
00:39:28,480 --> 00:39:30,890
OK, fine.

636
00:39:30,890 --> 00:39:34,090
So that's what
you have to solve.

637
00:39:34,090 --> 00:39:36,760
And actually I think of that
as the fundamental problem

638
00:39:36,760 --> 00:39:38,310
of numerical analysis.

639
00:39:38,310 --> 00:39:40,170
How to solve that equation.

640
00:39:40,170 --> 00:39:44,010
More effort, more thinking
goes into that than probably

641
00:39:44,010 --> 00:39:46,560
any other single problem.

642
00:39:46,560 --> 00:39:48,150
In some form.

643
00:39:48,150 --> 00:39:52,030
OK, and here's some
part of that thinking.

644
00:39:52,030 --> 00:39:55,100
Part of that thinking, and
another important possibility,

645
00:39:55,100 --> 00:40:00,840
is to keep-- This was like the
one equation, the one field

646
00:40:00,840 --> 00:40:06,470
problem, this corresponds
to the displacement method.

647
00:40:06,470 --> 00:40:11,940
Can I use words that I'm
not going to use seriously

648
00:40:11,940 --> 00:40:17,120
for another few weeks?

649
00:40:17,120 --> 00:40:20,030
This would correspond to
the displacement method

650
00:40:20,030 --> 00:40:22,640
in finite elements.

651
00:40:22,640 --> 00:40:25,710
In FEM, FEM for
finite element method.

652
00:40:25,710 --> 00:40:27,880
That's the displacement
method, it's the method

653
00:40:27,880 --> 00:40:30,800
that that most people use.

654
00:40:30,800 --> 00:40:33,040
It's the standard method.

655
00:40:33,040 --> 00:40:33,910
OK.

656
00:40:33,910 --> 00:40:36,410
But it's not the
only possibility,

657
00:40:36,410 --> 00:40:40,860
and let me show you a second
one that involves two equations.

658
00:40:40,860 --> 00:40:44,380
Because that's also
very important,

659
00:40:44,380 --> 00:40:47,860
with many other applications,
that we will see but

660
00:40:47,860 --> 00:40:49,110
haven't seen yet.

661
00:40:49,110 --> 00:40:54,320
So my two equations, I really
should say two systems,

662
00:40:54,320 --> 00:40:58,330
because one equation, that's
a vector equation of course.

663
00:40:58,330 --> 00:41:01,480
So I have a system of two
vector equations, that

664
00:41:01,480 --> 00:41:04,820
would go into a block
matrix and you'll see it.

665
00:41:04,820 --> 00:41:11,170
OK, so what what two
unknowns am I going to keep?

666
00:41:11,170 --> 00:41:12,880
u, I'm keeping.

667
00:41:12,880 --> 00:41:14,530
Displacement, I'm keeping.

668
00:41:14,530 --> 00:41:16,770
But I'm also going
to keep what I

669
00:41:16,770 --> 00:41:20,120
think of as the other
important unknown, w.

670
00:41:20,120 --> 00:41:22,500
So the other important
unknown is w.

671
00:41:22,500 --> 00:41:25,810
So now I have two
equations and one of them

672
00:41:25,810 --> 00:41:28,700
is just that, is
just the current,

673
00:41:28,700 --> 00:41:32,110
is just the current law, the
balance law A transpose w equal

674
00:41:32,110 --> 00:41:35,610
f.

675
00:41:35,610 --> 00:41:39,480
The only guy I'm
eliminating is e.

676
00:41:39,480 --> 00:41:45,910
Initially you could say I've a
three field system. u, e and w.

677
00:41:45,910 --> 00:41:50,090
Now, e and w are
so easily connected

678
00:41:50,090 --> 00:41:53,850
that I'm going to eliminate
e. e is c inverse w.

679
00:41:53,850 --> 00:41:56,240
I did it actually here.

680
00:41:56,240 --> 00:42:00,610
This w is C(b-Au),
that's we know.

681
00:42:00,610 --> 00:42:06,680
If I multiply by the C inverse,
then I have C inverse w is,

682
00:42:06,680 --> 00:42:11,900
and I bring the Au over to the
far left and I have only the b

683
00:42:11,900 --> 00:42:12,670
left.

684
00:42:12,670 --> 00:42:19,200
Everybody saw that, I
did a C inverse there

685
00:42:19,200 --> 00:42:26,330
to get e off by itself and
then I substituted for e,

686
00:42:26,330 --> 00:42:29,420
I put in the u part
so e's now gone,

687
00:42:29,420 --> 00:42:40,540
and the equation is C
inverse w plus Au equals b.

688
00:42:40,540 --> 00:42:46,090
That's my two field system.

689
00:42:46,090 --> 00:42:50,920
Now, there's a matrix here.

690
00:42:50,920 --> 00:42:53,187
This is really nice.

691
00:42:53,187 --> 00:42:54,270
I just want to write that.

692
00:42:54,270 --> 00:42:57,520
I've got to what I want but
now I want to look at it.

693
00:42:57,520 --> 00:43:04,190
So I think of a two by two
block matrix multiplying [w, u]

694
00:43:04,190 --> 00:43:07,590
and giving me [b, f].

695
00:43:07,590 --> 00:43:10,460
And I want to ask you
about that matrix.

696
00:43:10,460 --> 00:43:17,290
So this is the matrix for
when I've only eliminated e

697
00:43:17,290 --> 00:43:21,340
and I've still got
w as well as u.

698
00:43:21,340 --> 00:43:26,850
OK, you can read off
what's in that matrix.

699
00:43:26,850 --> 00:43:28,870
What goes up here?

700
00:43:28,870 --> 00:43:30,650
C inverse, of course.

701
00:43:30,650 --> 00:43:33,210
Positive diagonal, usually.

702
00:43:33,210 --> 00:43:34,060
Easy.

703
00:43:34,060 --> 00:43:37,980
What goes here is a rectangular,
that guy is rectangular.

704
00:43:37,980 --> 00:43:41,830
A transpose is multiplying
w, so it goes down here,

705
00:43:41,830 --> 00:43:47,750
and this equation
has no u in it.

706
00:43:47,750 --> 00:43:55,190
That matrix is worth noticing.

707
00:43:55,190 --> 00:43:58,410
And let's spend the rest of
this, the remaining minutes,

708
00:43:58,410 --> 00:44:00,320
just to think about that matrix.

709
00:44:00,320 --> 00:44:05,930
I just want to say,
what if I keep w and u,

710
00:44:05,930 --> 00:44:08,710
this is an important
possibility.

711
00:44:08,710 --> 00:44:12,820
And it's important in finite
elements which as you know

712
00:44:12,820 --> 00:44:17,010
is just a terrific way to
solve a whole lot of continuum

713
00:44:17,010 --> 00:44:18,100
problems.

714
00:44:18,100 --> 00:44:23,470
And what's it called when
I have w and u together,

715
00:44:23,470 --> 00:44:27,040
both unknowns, not
eliminating w now,

716
00:44:27,040 --> 00:44:29,020
it's called the mixed method.

717
00:44:29,020 --> 00:44:36,580
So this corresponds to the
mixed method in finite elements.

718
00:44:36,580 --> 00:44:45,420
It corresponds to the
possibility of keeping w and u.

719
00:44:45,420 --> 00:44:47,490
Well, and you might
say, wait, isn't there

720
00:44:47,490 --> 00:44:50,690
a third possibility?

721
00:44:50,690 --> 00:44:52,670
And what would that be?

722
00:44:52,670 --> 00:44:56,430
Keep only w.

723
00:44:56,430 --> 00:44:59,670
Here we kept only u,
here we've got them both,

724
00:44:59,670 --> 00:45:07,570
this is kind of the mother
of all equilibrium equations.

725
00:45:07,570 --> 00:45:10,570
And another possibly
would be to keep

726
00:45:10,570 --> 00:45:14,670
only the w's, to make the
currents the primary unknowns.

727
00:45:14,670 --> 00:45:19,620
And that, in the finite
element structural context,

728
00:45:19,620 --> 00:45:23,590
that would be saying
make the stresses.

729
00:45:23,590 --> 00:45:25,885
So of course it'd be
called the stress method.

730
00:45:25,885 --> 00:45:27,260
It'd be called
the stress method,

731
00:45:27,260 --> 00:45:31,660
and Professor Pian in
Course 16, now retired,

732
00:45:31,660 --> 00:45:35,000
was one of the major developers
of the stress method.

733
00:45:35,000 --> 00:45:36,960
The difficulty with
the stress method,

734
00:45:36,960 --> 00:45:41,330
the reason it
didn't win big time,

735
00:45:41,330 --> 00:45:46,375
is that the w's, if you make
them the unknowns you've got

736
00:45:46,375 --> 00:45:49,970
a constraint on them,
Kirchhoff's-- Not all w's are

737
00:45:49,970 --> 00:45:52,230
allowed.

738
00:45:52,230 --> 00:45:55,510
Somehow over here
all u's are allowed,

739
00:45:55,510 --> 00:46:00,830
and that made it much easier
to set up the problem.

740
00:46:00,830 --> 00:46:05,320
So the displacement method
is the 95 percent winner.

741
00:46:05,320 --> 00:46:10,400
But there are problems where
maybe C inverse is complicated,

742
00:46:10,400 --> 00:46:12,530
or C is too
complicated and you're

743
00:46:12,530 --> 00:46:18,540
better to-- We can see that.

744
00:46:18,540 --> 00:46:21,470
That's later in the
book, but we want

745
00:46:21,470 --> 00:46:24,400
to see now about that matrix.

746
00:46:24,400 --> 00:46:27,970
Well, if I wrote
that matrix down,

747
00:46:27,970 --> 00:46:32,080
and let me write just so
you-- I want to ask you

748
00:46:32,080 --> 00:46:34,070
about that block matrix.

749
00:46:34,070 --> 00:46:36,840
What's its size?

750
00:46:36,840 --> 00:46:39,260
Now just focus entirely
on that block matrix,

751
00:46:39,260 --> 00:46:41,460
because that's
what I care about.

752
00:46:41,460 --> 00:46:46,300
What's the size of that matrix?

753
00:46:46,300 --> 00:46:50,790
Let's see, what's the
size of C? m by m.

754
00:46:50,790 --> 00:46:53,740
What's the size of A? n by n.

755
00:46:53,740 --> 00:46:54,840
So what do I have here?

756
00:46:54,840 --> 00:46:58,470
I've got n rows and
m plus n columns,

757
00:46:58,470 --> 00:46:59,680
and there's n more rows.

758
00:46:59,680 --> 00:47:01,310
It's of size m+n.

759
00:47:01,310 --> 00:47:05,880


760
00:47:05,880 --> 00:47:08,760
It's got the n
u's and the m w's.

761
00:47:08,760 --> 00:47:10,910
Of course, m+n.

762
00:47:10,910 --> 00:47:12,890
And the natural size, right.

763
00:47:12,890 --> 00:47:15,500
So it's got more
unknowns but we'll

764
00:47:15,500 --> 00:47:17,880
see, oh in
optimization you bring

765
00:47:17,880 --> 00:47:22,050
in Lagrange multipliers, that's
just exactly parallel to what

766
00:47:22,050 --> 00:47:23,070
we're doing here.

767
00:47:23,070 --> 00:47:26,080
You have more, you have
extra bunch of unknowns.

768
00:47:26,080 --> 00:47:27,460
That's what we have.

769
00:47:27,460 --> 00:47:32,190
Now what else about that matrix?

770
00:47:32,190 --> 00:47:35,880
I was going to write down
a very, very tiny model

771
00:47:35,880 --> 00:47:37,800
for that matrix.

772
00:47:37,800 --> 00:47:39,220
I'll just make it two by two.

773
00:47:39,220 --> 00:47:44,830
Here's a model for that
matrix where C is just a one

774
00:47:44,830 --> 00:47:48,560
and A is just a one.

775
00:47:48,560 --> 00:47:51,000
I mean, it's kind
of laughable, right?

776
00:47:51,000 --> 00:47:53,190
That model, this
is the real thing.

777
00:47:53,190 --> 00:48:00,330
But it gives you an
example to check against.

778
00:48:00,330 --> 00:48:04,250
OK, what's a property
of that matrix?

779
00:48:04,250 --> 00:48:06,070
It's, again?

780
00:48:06,070 --> 00:48:06,880
Symmetric.

781
00:48:06,880 --> 00:48:07,610
Good.

782
00:48:07,610 --> 00:48:09,820
That's a symmetric matrix.

783
00:48:09,820 --> 00:48:13,360
Because what happens if I
transpose that block matrix?

784
00:48:13,360 --> 00:48:18,700
That A block will flip
over here as A transpose,

785
00:48:18,700 --> 00:48:21,310
the A transpose block
will flip up there

786
00:48:21,310 --> 00:48:26,050
as A, what happens to
the C inverse block?

787
00:48:26,050 --> 00:48:28,520
C is a symmetric guy.

788
00:48:28,520 --> 00:48:32,900
In fact, it was just
diagonal in our imagination.

789
00:48:32,900 --> 00:48:37,110
The key point is it's symmetric,
its inverse is symmetric,

790
00:48:37,110 --> 00:48:39,320
its transpose is the same.

791
00:48:39,320 --> 00:48:42,200
So that's a symmetric matrix.

792
00:48:42,200 --> 00:48:43,450
That's a good thing, right?

793
00:48:43,450 --> 00:48:49,270
Now we've got a matrix that's
symmetric, square symmetric.

794
00:48:49,270 --> 00:48:53,490
OK, what's my other
question about that matrix?

795
00:48:53,490 --> 00:49:00,570
Is it or is it not
positive definite, right?

796
00:49:00,570 --> 00:49:03,510
We've got to answer
that question.

797
00:49:03,510 --> 00:49:06,040
Have we got a positive
definite matrix?

798
00:49:06,040 --> 00:49:08,830
Would all the
pivots be positive?

799
00:49:08,830 --> 00:49:11,990
Would the eigenvalues
be positive?

800
00:49:11,990 --> 00:49:15,970
What's your guess?

801
00:49:15,970 --> 00:49:16,790
No.

802
00:49:16,790 --> 00:49:19,160
That matrix is not positive.

803
00:49:19,160 --> 00:49:22,440
No way that a matrix
with a zero there,

804
00:49:22,440 --> 00:49:25,740
a zero block, or that
matrix with a zero number

805
00:49:25,740 --> 00:49:27,060
could be positive definite.

806
00:49:27,060 --> 00:49:28,210
No, no way.

807
00:49:28,210 --> 00:49:39,770
The energy in this guy, this
u transpose Au, you remember,

808
00:49:39,770 --> 00:49:46,280
would be u_1 squared
and u_1*u_2, twice.

809
00:49:46,280 --> 00:49:50,040
And no u_2 squareds.

810
00:49:50,040 --> 00:49:55,300
And that thing is
definitely indefinite.

811
00:49:55,300 --> 00:49:59,050
Right?

812
00:49:59,050 --> 00:50:01,840
In the u_1 direction
it looks good,

813
00:50:01,840 --> 00:50:03,800
that's things positive there.

814
00:50:03,800 --> 00:50:08,070
But if I took u_1 and u_2
to have opposite signs,

815
00:50:08,070 --> 00:50:11,470
and made u_2 big
enough then of course

816
00:50:11,470 --> 00:50:12,650
this just brings it down.

817
00:50:12,650 --> 00:50:16,030
So the graph of that
would be a saddle point.

818
00:50:16,030 --> 00:50:18,170
The graph of that
would be a saddle.

819
00:50:18,170 --> 00:50:26,040
OK, and now here I have the
same thing on an m+n size.

820
00:50:26,040 --> 00:50:26,920
So what do I have?

821
00:50:26,920 --> 00:50:28,890
Actually, you could see.

822
00:50:28,890 --> 00:50:34,120
The last exercise is mentally
do elimination on that matrix.

823
00:50:34,120 --> 00:50:37,160
Mentally do elimination
on that matrix.

824
00:50:37,160 --> 00:50:42,910
So start with the first m rows.

825
00:50:42,910 --> 00:50:45,000
We'll work with those first.

826
00:50:45,000 --> 00:50:48,090
What will elimination do,
what will the pivots be like,

827
00:50:48,090 --> 00:50:51,460
what will happen at the
beginning of elimination?

828
00:50:51,460 --> 00:50:55,790
When I start with this matrix?

829
00:50:55,790 --> 00:50:59,430
Well it meets C inverse right
away, that diagonal matrix,

830
00:50:59,430 --> 00:51:00,850
and it's extremely happy.

831
00:51:00,850 --> 00:51:02,300
Those will be the pivots, right?

832
00:51:02,300 --> 00:51:03,740
They're sitting
on the diagonals,

833
00:51:03,740 --> 00:51:05,510
zero off the diagonals.

834
00:51:05,510 --> 00:51:09,460
They'll be positive pivots, I'll
have m positive pivots here.

835
00:51:09,460 --> 00:51:15,740
And then I get down to where
A comes in the picture.

836
00:51:15,740 --> 00:51:20,800
So on the last board here,
let me just copy this matrix,

837
00:51:20,800 --> 00:51:25,320
[C inverse, A; A transpose, 0].

838
00:51:25,320 --> 00:51:29,210
An elimination is going to,
it'll be very happy with that.

839
00:51:29,210 --> 00:51:35,570
But it's going to put, so
it's happy with that row.

840
00:51:35,570 --> 00:51:36,660
Block row.

841
00:51:36,660 --> 00:51:41,130
It's going to do an elimination
to get a bunch of zeroes there

842
00:51:41,130 --> 00:51:45,740
and what did it do?

843
00:51:45,740 --> 00:51:48,310
This was elimination,
this was subtracting,

844
00:51:48,310 --> 00:51:50,700
yeah what did it subtract here?

845
00:51:50,700 --> 00:51:55,080
It multiplied these
pivot rows by something

846
00:51:55,080 --> 00:52:01,270
and subtracted from these lower
rows and got the zero block.

847
00:52:01,270 --> 00:52:04,130
And what did it multiply by?

848
00:52:04,130 --> 00:52:07,480
What do I multiply that
block row, and this

849
00:52:07,480 --> 00:52:11,730
is a perfect, perfect exercise
to see how blocks are just

850
00:52:11,730 --> 00:52:12,880
like numbers.

851
00:52:12,880 --> 00:52:14,110
You can deal with them.

852
00:52:14,110 --> 00:52:17,230
What do I multiply
that block row by

853
00:52:17,230 --> 00:52:22,880
and subtract from the
row below to get a zero.

854
00:52:22,880 --> 00:52:27,090
You said C A transpose, but
I don't think that's it.

855
00:52:27,090 --> 00:52:31,100
A transpose C. You've got to
multiply by A transpose C.

856
00:52:31,100 --> 00:52:34,690
First of all, C A transpose
wouldn't be a possibility.

857
00:52:34,690 --> 00:52:35,970
Wrong shapes.

858
00:52:35,970 --> 00:52:42,210
A transpose C is the four
by five, five by five guy.

859
00:52:42,210 --> 00:52:45,470
So you multiply A transpose
C, that cancels that,

860
00:52:45,470 --> 00:52:48,720
leaves the A transpose, when you
subtract it gives you a zero,

861
00:52:48,720 --> 00:52:51,100
and what does it give you there?

862
00:52:51,100 --> 00:52:55,310
What shows up there?

863
00:52:55,310 --> 00:53:00,710
A transpose C multiplies
at A, subtracts

864
00:53:00,710 --> 00:53:04,780
so it's actually what shows up
there is minus A transpose C

865
00:53:04,780 --> 00:53:08,200
A. Let me write it in there.

866
00:53:08,200 --> 00:53:11,980
Minus A transpose
C A. So that matrix

867
00:53:11,980 --> 00:53:13,880
is exactly what
comes from this one.

868
00:53:13,880 --> 00:53:16,870
It's exactly what we
do when we eliminate w.

869
00:53:16,870 --> 00:53:18,250
That's what elimination is.

870
00:53:18,250 --> 00:53:21,400
I just eliminated w by
getting a zero there.

871
00:53:21,400 --> 00:53:25,500
And I got only an equation,
but notice the minus.

872
00:53:25,500 --> 00:53:32,030
So, final question, what are
the signs of the last n pivots?

873
00:53:32,030 --> 00:53:35,370
The first m were all positive,
and they were sitting

874
00:53:35,370 --> 00:53:38,120
on the diagonal already.

875
00:53:38,120 --> 00:53:42,430
The last n are not
so easy to see,

876
00:53:42,430 --> 00:53:44,860
but we can see what
sign they have.

877
00:53:44,860 --> 00:53:48,390
And what sign do the
last n pivots have?

878
00:53:48,390 --> 00:53:50,090
Minus.

879
00:53:50,090 --> 00:53:53,160
Because they come from
a negative definite.

880
00:53:53,160 --> 00:53:55,850
Minus A transpose C
A is shown up there.

881
00:53:55,850 --> 00:53:58,630
So that's the saddle point.

882
00:53:58,630 --> 00:54:03,230
Saddle points are, when you
have two-field problems,

883
00:54:03,230 --> 00:54:05,680
you're talking
about saddle points,

884
00:54:05,680 --> 00:54:09,590
and the mixed method in finite
elements is exactly that.

885
00:54:09,590 --> 00:54:17,420
And the tricky part is
then with the mixed method,

886
00:54:17,420 --> 00:54:19,800
you're sort of not so
perfectly guaranteed

887
00:54:19,800 --> 00:54:21,980
that the matrix is invertible.

888
00:54:21,980 --> 00:54:26,210
Because we have plus
stuff and minus stuff.

889
00:54:26,210 --> 00:54:28,620
OK, thank you, that's great.

890
00:54:28,620 --> 00:54:33,830
And I'll see you Monday all
about the exam and review,

891
00:54:33,830 --> 00:54:36,310
it's a great chance
to think back.