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GILBERT STRANG: OK.

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So this video is about using
eigenvectors and eigenvalues

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to take powers of
a matrix, and I'll

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show you why we want to
take powers of a matrix.

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And then the next
video would be using

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eigenvalues and eigenvectors to
solve differential equations.

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The two big applications.

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So here's the first application.

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Let me remember the main facts.

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That if A-- if.

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This is an important point.

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Not every matrix has n
independent eigenvectors that

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would go into matrix
V. You remember

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V is the eigenvector
matrix, and I

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need n independent eigenvectors
in order to have a V inverse,

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to make that formula correct.

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So that's the key
formula for using

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eigenvalues and eigenvectors.

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And the case where we might
run short of eigenvectors

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is when maybe one
eigenvalue is repeated.

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It's a double eigenvalue,
and maybe there's

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only one eigenvector
to go with it.

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Every eigenvalue's got at
least one line of eigenvectors.

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But we might not have two when
the eigenvalue is repeated

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or we might.

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So there are cases when
this formula doesn't apply.

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Because I must be able
to take V inverse,

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I need n independent
columns there.

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OK.

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But when it works,
it really works.

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So the n-th power,
just remembering,

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is V lambda V inverse, V
lambda V inverse, n times.

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But every time I have
a V inverse and a V,

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that's the identity.

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So I move V out
at the beginning.

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I have lambda, lambda,
lambda, n of those,

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and a V inverse at the very end.

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So that's the nice result for
the n-th power of a matrix.

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Now I have to show you
how to use that formula,

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how to use eigenvalues
and eigenvectors.

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OK.

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So we know we can take
powers of a matrix.

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So first of all, what
kind of equation?

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There's an equation.

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That's called a
difference equation.

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It goes from step k to step
k plus 1 to step k plus 2.

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It steps one at a time and
every time multiplies by A.

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So after k steps, I've
multiplied by A k times

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from the original u0.

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So instead of a
differential equation,

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it's a step difference
equation with u0 given.

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And there's the solution.

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That's the quickest
form of the solution.

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A to the k-th power,
that's what we want.

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But just writing
A to the k, if we

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had a big matrix, to
take its hundredth power

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would be ridiculous.

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But with eigenvalues
and eigenvectors,

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we have that formula.

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OK.

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But now I want to think.

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Let me try to turn that
formula into something

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that you just naturally see.

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And we know what happens.

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If u0 is an eigenvector,
if u0 is an eigenvector,

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that probably won't happen
because there are just

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n eigenvector directions.

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But if it happened to be an
eigenvector, then every step

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we'd multiply by lambda, and
we'd have the answer, lambda k

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times.

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But what do we do for all
the initial vectors u0 which

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are maybe not an eigenvector?

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How do I proceed?

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How do I use eigenvectors when
my original starting vector

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is not an eigenvector?

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And the answer is, it will be
a combination of eigenvectors.

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So making this formula
real starts with this.

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So I write u0 as a combination
of the eigenvectors.

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And I can do it
because if I have

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n independent eigenvectors,
that will be a basis.

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Every vector can be
written in the basis.

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So I'm looking there at a
combination of eigenvectors.

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And now the point is that as
I take these steps to u1--

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what will u1 be?

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u1 will be Au0.

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So I'm multiplying by A. So
when I multiply this by A,

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what happens?

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That's the whole point.

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c1, A times x1 is
lambda 1 times x1.

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It's an eigenvector.

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c2 tells me how much of the
second eigenvector I have.

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When I multiply by A, that
multiplies by lambda 2,

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and so on, cn lambda n xn.

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And that's the thing.

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Each eigenvector
goes its own way,

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and I just add them together.

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OK.

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And what about A
to the k-th power?

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Now, that will give me uk.

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And what happens if
I do this k times?

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You've seen what I got
after doing it one time.

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If I do it k times,
that lambda 1

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that multiplies its eigenvector
will happen k times.

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So I'll have lambda
1 to the k-th power.

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Do you see that?

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Every step brings
another factor lambda 1.

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Every step brings
another factor lambda 2.

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Every step brings--
that's the answer.

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That is-- well, that answer
must be the same as this answer.

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And I'll do an
example in a minute.

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Right now, I'm just getting
the formulas straight.

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So I have the quickest
possible formula,

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but it doesn't help me much.

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I have the using the
eigenvectors and eigenvalue

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formula.

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And here I have it
that, really, it's

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the same thing written out as
a combination of eigenvectors.

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And then this is my answer.

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That's my answer to the--
that's my solution uk.

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That's it.

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So that must be
the same as that.

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Do you want to just
think for one minute

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why this answer is the
same as that answer?

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Well, we need to know
what are the c's?

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Well, the c's came from u0.

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And if I write that equation
for the c's-- do you see what I

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have as an equation for the c's?

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u0 is this combination
of eigenvectors.

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That's a matrix multiplication.

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That's the eigenvector matrix
multiplied by the vector

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c of coefficients, right?

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That's how a matrix
multiplies a vector.

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The columns, which are the x's,
multiply the numbers c1, c2,

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cn.

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There it is.

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That's the same as that.

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So u0 is Vc.

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So c is V inverse u0.

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Oh, that's nice.

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That's telling us what
are the coefficients, what

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are the numbers, what
amount of each eigenvector

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is present in u0.

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This is the equation.

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But look, you see
there that V inverse

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u0, that's the first part
there of the formula.

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I'm trying to match this
formula with that one.

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And I'm taking one
step to recognize

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that this part of the
formula is exactly c.

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You might want to
think about that.

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Run this video once more
just to see that step.

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Now what do we do?

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We've got the lambdas.

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So I'm taking care of
the c's, you could say.

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Now I need the lambda
to the k-th power--

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lambda 1 to the k-th, lambda
2 to the k-th, lambda n

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to the k-th.

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That's exactly
what goes in here.

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So that factor is producing
the lambdas to the k-th power.

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And finally, this factor has--
everybody's remembering here.

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V is the eigenvector
matrix x1, x2, to xn.

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And when I multiply by V,
it's a matrix times a vector.

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This is a matrix.

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This is a vector.

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And I get the combination--
I'm adding up.

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I'm reconstructing the solution.

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So first I break
up u0 into the x's.

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I multiply them by the
lambdas, and then I

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put them all together.

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I reconstruct uk.

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I hope you like that.

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This formula, which it's
like common sense formula,

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is exactly what that algebra
formula, matrix formula, says.

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OK.

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I have to do an example.

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Let me finish with an example.

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OK.

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Here's a matrix example.

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A equals-- this'll
be a special matrix.

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I'm going to make the
first column add up to 1,

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and I'm going to make the
second column add up to 1.

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And I'm using positive numbers.

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They're adding to 1.

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And that's called
a Markov matrix.

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So it's nice to know that
name-- Markov matrix.

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One of the beauties
of linear algebra

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is the variety of matrices--
orthogonal matrices,

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symmetric matrices.

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We'll see more and
more kinds of matrices.

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And sometimes they're
named after somebody

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who understood that
they were important

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and found their
special properties.

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So a Markov matrix is a matrix
with the columns adding up

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to 1 and no negative numbers
involved, no negative numbers.

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OK.

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That's just by the way.

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But it tells us something
about the eigenvalues here.

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Well, we could find
those two eigenvalues.

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We could do the determinant.

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You remember how to
find eigenvalues.

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The determinant of lambda I
minus A will be something.

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Could easily figure it out.

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There's always a lambda squared,
because it's two by two,

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minus the trace.

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0.8 and 0.7 is 1.5 lambda,
plus the determinant.

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0.56 minus 0.06 is 0.50, 0.5.

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And you set that to 0.

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And you get a result that
one of the eigenvalues

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is-- this factors into lambda
minus 1, lambda minus 1/2.

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And the cool fact
about Markov matrices

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is lambda equal 1 is
always an eigenvalue.

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So lambda equal 1
is an eigenvalue.

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Let's call that lambda 1.

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And lambda 2 is an
eigenvalue, and that

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depends on the numbers,
and it's 1/2, 0.5, 0.5.

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Those are the eigenvalues.

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1 plus 1/2 is 1.5.

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The trace is 0.8 plus 0.7, 1.5.

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Are we good for those
two eigenvalues?

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Yes.

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And then we find the
eigenvectors that go with them.

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I think that this eigenvector
turns out to be 0.6, 0.4.

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I could check.

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If I multiply, I get
0.48 plus 0.12 is 0.60,

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and that's the same as 0.6.

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And that goes with eigenvalue 1.

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And I think that this
eigenvector is 1, minus 1.

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Maybe that's always for a
two-by-two Markov matrix.

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Maybe that's always
the second eigenvector.

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I think that's probably good.

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Right.

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OK.

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Yeah.

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All right.

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What now?

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What now?

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I want to use the
eigenvalues and eigenvectors,

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and I'm going to
write out now uk.

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So if I apply A k
times to u0, I get uk.

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And that's c1 1 to
the k-- this lambda 1

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is 1-- times its eigenvector
0.6, 0.4 plus c2,

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however much of the
second eigenvector

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is in there, times
its eigenvalue,

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1/2 to the k-th power
times its eigenvector,

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the second eigenvector,
1, negative 1.

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That is a formula.

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c1 lambda 1 to the k-th
power x1 plus c2 lambda 2

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00:14:56,985 --> 00:14:59,660
to the k-th power x2.

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And c1 and c2 would
be determined by u0,

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which I haven't picked a u0.

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I could.

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But I can make the
point, because the point

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I want to make is true for
every u0, every example.

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And here's the point.

254
00:15:20,410 --> 00:15:22,760
What happens as k gets large?

255
00:15:22,760 --> 00:15:27,500
What happens if Markov
multiplies his matrix over

256
00:15:27,500 --> 00:15:32,000
and over again, which is what
happens in a Markov process,

257
00:15:32,000 --> 00:15:33,770
a Markov process?

258
00:15:33,770 --> 00:15:37,800
This is like-- actually,
the whole Google algorithm

259
00:15:37,800 --> 00:15:42,610
for page rank is based
on a Markov matrix.

260
00:15:42,610 --> 00:15:46,540
So that's like a
multi-billion-dollar company

261
00:15:46,540 --> 00:15:52,240
that's based on the
properties of a Markov matrix.

262
00:15:52,240 --> 00:15:55,060
And you repeat it and repeat it.

263
00:15:55,060 --> 00:16:00,500
That just means that Google
is looping through the web,

264
00:16:00,500 --> 00:16:06,680
and if it sees a website more
often, the ranking goes up.

265
00:16:06,680 --> 00:16:09,670
And if it never sees
my website, then

266
00:16:09,670 --> 00:16:14,240
for that, when it was
googling some special subject,

267
00:16:14,240 --> 00:16:20,080
it never came to your website
and mine, we didn't get ranked.

268
00:16:20,080 --> 00:16:20,870
OK.

269
00:16:20,870 --> 00:16:22,840
So this goes to 0.

270
00:16:22,840 --> 00:16:26,590
1/2 to the-- it goes
fast to 0, quickly to 0.

271
00:16:26,590 --> 00:16:28,870
So that goes to 0.

272
00:16:28,870 --> 00:16:31,900
And of course, that stays
exactly where it is.

273
00:16:31,900 --> 00:16:34,870
So there's a steady state.

274
00:16:34,870 --> 00:16:40,500
What happens if page rank had
only two websites to rank,

275
00:16:40,500 --> 00:16:43,270
if Google was just
ranking two websites?

276
00:16:43,270 --> 00:16:48,000
Then its initial ranking,
they don't know what it is.

277
00:16:48,000 --> 00:16:51,980
But by repeating the
Markov matrix and this part

278
00:16:51,980 --> 00:16:55,590
going to 0, right, goes
to 0 because of 1/2

279
00:16:55,590 --> 00:17:00,530
to the k-th power, there
is the ranking, 0.6, 0.4.

280
00:17:00,530 --> 00:17:04,130
That's where Google-- so
this first website would

281
00:17:04,130 --> 00:17:06,980
be ranked above the second one.

282
00:17:06,980 --> 00:17:07,530
OK.

283
00:17:07,530 --> 00:17:12,180
There's an example of a process
that's repeated and repeated,

284
00:17:12,180 --> 00:17:16,910
and so a Markov matrix comes in.

285
00:17:16,910 --> 00:17:20,599
This business of adding up to
1 means that nothing is lost.

286
00:17:20,599 --> 00:17:21,760
Nothing is created.

287
00:17:21,760 --> 00:17:23,130
You're just moving.

288
00:17:23,130 --> 00:17:26,550
At every step, you
take a Markov step.

289
00:17:26,550 --> 00:17:29,420
And the question is,
where do you end up?

290
00:17:29,420 --> 00:17:32,950
Well, you keep moving,
but this vector

291
00:17:32,950 --> 00:17:36,060
tells you how much of
the time you're spending

292
00:17:36,060 --> 00:17:40,905
in the two possible locations.

293
00:17:40,905 --> 00:17:43,100
And this one goes to 0.

294
00:17:43,100 --> 00:17:43,940
OK.

295
00:17:43,940 --> 00:17:47,450
Powers of a matrix,
powers of a Markov matrix.

296
00:17:47,450 --> 00:17:49,230
Thank you.