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GILBERT STRANG: OK, thanks.

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Here's a second video
that involves the matrix

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exponential.

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But it has a new idea
in it, a basic new idea.

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And that idea is two matrices
being called "similar."

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So that word "similar"
has a specific meaning,

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that a matrix A, is
similar to another matrix

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B, if B comes from A this way.

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Notice this way.

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It means there's some matrix M--
could be any invertible matrix.

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So that I take A,
multiply on the right

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by M and on the
left by M inverse.

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That'd probably give
me a new matrix.

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Call it B. That matrix
is called "similar" to B.

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I'll show you examples of
matrices that are similar.

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But first is to get
this definition in mind.

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So in general, a
lot of matrices are

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similar to-- if I have a certain
matrix A, I can take any M,

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and I'll get a similar
matrix B. So there

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are lots of similar matrices.

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And the point is all
those similar matrices

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have the same eigenvalues.

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So there's a little
family of matrices

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there, all similar
to each other and all

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with the same eigenvalues.

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Why do they have the
same eigenvalues?

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I'll just show you, one line.

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Suppose B has an
eigenvalue of lambda.

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So B is M inverse AM.

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So I have this.

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M inverse AMx is lambda x.

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That's Bx.

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B has an eigenvalue of lambda.

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I want to show that A has
an eigenvalue of lambda.

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OK.

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So I look at this.

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I multiply both sides
by M. That cancels this.

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So when I multiply by M,
this is gone, and I have AMx.

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But the M shows up on
the right-hand side,

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I have lambda Mx.

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Now I would just look
at that, and I say, yes.

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A has an eigenvector-- Mx
with eigenvalue lambda.

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A times that vector is
lambda times that vector.

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So lambda is an eigenvalue of A.
It has a different eigenvector,

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of course.

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If matrices have
the same eigenvalues

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and the same eigenvectors,
that's the same matrix.

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But if I do this,
allow an M matrix

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to get in there, that
changes the eigenvectors.

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Here they were
originally x for B.

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And now for A,
they're M times x.

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It does not change the
eigenvalues because of this M

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on both sides allowed
me to bring M over

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to the right-hand side
and make that work.

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OK.

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Here are some similar matrices.

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Let me take some.

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So these will be all similar.

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Say 2, 3, 0, 4.

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OK?

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That's a matrix A. I can see
its eigenvalues are 2 and 4.

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Well, I know that it will be
similar to the diagonal matrix.

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So there is some matrix
M that connects this one

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with this one, connects
this A with that B. Well,

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that B is really capital lambda.

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And we know what matrix
connects the original A

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to its eigenvalue matrix.

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What is the M that does that?

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It's the eigenvector matrix.

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So to get this particular--
to get this guy,

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starting from here, I use
M is V for this example

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to produce that.

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Then B is lambda.

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But there are other
possibilities.

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So let me see.

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I think probably a
matrix is-- there

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is the matrix, A transpose.

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Is that similar to A?

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Is A transpose similar to A?

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Well, answer-- yes.

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The transpose matrix has those
same eigenvalues, 2 and 4,

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and different eigenvectors.

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And those eigenvectors
would connect the original A

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and this A or that A transpose.

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So the transpose of a matrix
is similar to the matrix.

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What about if I
change the order?

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4, 0, 0, 2.

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So I've just flipped
the 2 and the 4,

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but of course I haven't
changed the eigenvalues.

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You could find the
M that does that.

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You can find an M so that
if I multiply on the right

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by M and on the left by M
inverse, it flips those.

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So there's another
matrix similar.

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Oh, there could be plenty more.

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All I want to do is have
the eigenvalues be 4 and 2.

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Shall I just create some more?

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Here is a 0, 6.

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I wanted to get the trace right.

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4 plus 2 matches 0 plus 6.

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Now I have to get the
determinant right.

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That has a determinant of 8.

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What about a 2 and
a minus 4 there?

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I think I've got the
trace correct-- 6.

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And I've got the
determinant correct-- 8.

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And there the determinant is 8.

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So that would be
a similar matrix.

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All similar matrices.

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A family of similar matrices
with the eigenvalues 4 and 2.

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So I want to do another
example of similar matrices.

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What will be different
in this example

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is there'll be
missing eigenvectors.

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So let me say, 2, 2, 0, 1.

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So that has eigenvalues 2 and
2 but only one eigenvector.

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Here is another
matrix like that.

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Say, so the trace should be 4.

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The determinant should be 4.

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So maybe I put a 2
and a minus 2 there.

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I think that has the
correct trace, 4,

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and the great
determinant, also 4.

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So that will have eigenvalues 2
and 2 and only one eigenvector,

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so it's similar to this.

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Now here's the point.

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You might say, what
about 2, 2, 0, 0.

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That has the
correct eigenvalues,

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but it's not similar.

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There's no matrix M that
connects that diagonal matrix

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with these other matrices.

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That matrix has no
missing eigenvectors.

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These matrices have one
missing eigenvector.

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What's called the Jordan form.

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The Jordan form.

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So that didn't belong.

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That's not in that family.

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The Jordan form is--
you could say-- well,

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that'll be the Jordan form.

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The most beautiful member of
the family is the Jordan form.

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So I have a whole lot of
matrices that are similar.

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That is the most beautiful,
but it's not in the family.

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It's related but
not in the family.

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It's not similar to those.

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And the best one
would be this one.

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So the Jordan form would be
that one with the eigenvalues

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on the diagonal.

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But because there's a
missing eigenvector,

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there has to be a
reason for that.

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And it's in the 1 there,
and I can't have a 0 there.

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OK.

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So that's the idea
of similar matrices.

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And now I do have one more
important note, a caution

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about matrix exponentials.

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Can I just tell you this
caution, this caution?

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If I look at e to the
A times e to the B.

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The exponential of A times
the exponential of B.

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My caution is that usually that
is not e to the B, e to the A.

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If I put B and A in
the opposite order,

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I get something different.

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And it's also not
e to the A plus B.

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Those are all different.

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Which, if I had 1 by 1, just
numbers here, of course,

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that's the great rule
for exponentials.

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But for matrix exponentials,
that rule doesn't work.

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That is not the same
as e to the A plus B.

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And I can show you why.

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e to the A is I plus A plus
1/2 A squared and so on.

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e to the B is I plus B plus
1/2 B squared and so on.

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And I do that multiplication.

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And I get I. And I get an
A. And I get a B times I.

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And now I get 1/2 B squared
and an AB and 1/2 A squared.

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Can I put those down?

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1/2 A squared, and
there's an A times a B.

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And there's a 1/2 B squared.

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OK.

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This makes the point.

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If I multiply the
exponentials in this order,

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I get A times B. What
if I multiply them

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in the other order,
in that order?

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If I multiply e to the
B times e to the A,

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then the B's will be
out in front of the A's.

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And this would become a
BA, which can be different.

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So already I see that
the two are different.

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Here is e to the
A, e to the B. It

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has A before B. If
I do it this way,

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it'll have B before A.
If I do it this way,

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it'll have a mixture.

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So e to the A plus B
will have a I and an A

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and a B and a 1/2
A plus B squared.

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So that'll be 1/2 of A squared
plus AB plus BA plus B squared.

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Different again.

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Now I have a sort of
symmetric mixture of A and B.

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In this case, I had A
before B. In this case,

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I had B on the left side of A.

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So all three of
those are different,

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even in this term
of the series that

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defines those exponentials.

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And that means that
systems of equations,

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if the coefficients change over
time, are definitely harder.

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We were able to solve
dy dt equals, say,

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cosine of t times y.

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Do you remember how-- that
this was solvable for a 1 by 1.

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We put the exponent--
the solution was y

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is e to the-- we
integrated cosine t

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and got sine t times y of 0.

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e to the sine t--

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Can I just think of putting
that into the differential

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equation-- its derivative.

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The derivative of e to the
sine t will be e to the sine t.

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I'm using the chain rule.

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The derivative of e to the
sine t will be e to the sine t

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again, times the derivative
of sine t, which is cos t,

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so it works.

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That's fine as a solution.

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But if I have matrices
here-- if I have matrices,

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then the whole thing goes wrong.

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You could say that the
chain rule goes wrong.

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You can't put the
integral up there

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and then take the derivative
and expect it to come back down.

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The chain rule will not work
for matrix exponentials,

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the simple chain rule.

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And the fact is
that we don't have

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nice formulas for the
solutions to linear systems

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00:14:06,640 --> 00:14:10,610
with time-varying coefficients.

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00:14:10,610 --> 00:14:15,600
That has become a harder problem
when we went from one equation

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to a system of an equation.

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So this is the caution slide
about matrix exponentials.

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They're beautiful.

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They work perfectly if you
just have one matrix A.

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But if somehow two
matrices are in there

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or a bunch of
different matrices,

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then you lose the good rules,
and you lose the solution.

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OK.

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Thank you.