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GILBERT STRANG: OK.
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More about eigenvalues
and eigenvectors.
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Well, actually, it's
going to be the same thing
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about eigenvalues
and eigenvectors
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but I'm going to
use matrix notation.
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So, you remember I have a
matrix A, 2 by 2 for example.
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It's got two eigenvectors.
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Each eigenvector
has its eigenvalue.
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So I could write the
eigenvalue world that way.
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I want to write
it in matrix form.
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I want to create an
eigenvector matrix
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by taking the two
eigenvectors and putting them
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in the columns of my matrix.
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If I have n of them, that
allows me to give one name.
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The eigenvector matrix, maybe
I'll call it V for vectors.
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So that's A times V. And
now, just bear with me
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while I do that
multiplication of A times
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the eigenvector matrix.
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So what do I get?
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I get a matrix.
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That's 2 by 2.
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That's 2 by 2.
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You get a 2 by 2 matrix.
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What's the first column?
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The first column of
the output is A times
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the first column of the input.
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And what is A times x1?
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Well, A times x1 is
lambda 1 times x1.
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So that first column
is lambda 1 x1.
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And A times the second column
is Ax2, which is lambda 2 x2.
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So I'm seeing lambda
2 x2 in that column.
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OK.
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Matrix notation.
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Those were the eigenvectors.
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This is the result of A times V.
But I can look at this a little
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differently.
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I can say, wait a minute, that
is my eigenvector matrix, x1
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and x2-- those two
columns-- times a matrix.
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Yes.
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Taking this first column, lambda
1 x1, is lambda 1 times x1,
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plus 0 times x2.
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Right there I did a
matrix multiplication.
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I did it without
preparing you for it.
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I'll go back and do that
preparation in a moment.
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But when I multiply a matrix by
a vector, I take lambda 1 times
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that one, 0 times that one.
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I get lambda 1 x1,
which is what I want.
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Can you see what I want
in the second column here?
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The result I want
is lambda 2 x2.
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So I want no x1's, and
lambda 2 of that column.
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So that's 0 times that
column, plus lambda 2,
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times that column.
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Are we OK?
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So, what do I have now?
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I have the whole thing
in a beautiful form,
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as this A times the
eigenvector matrix
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equals, there is the
eigenvector matrix again, V.
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And here is a new matrix
that's the eigenvalue matrix.
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And everybody calls
that-- because those
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are lambda 1 and lambda 2.
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So the natural letter
is a capital lambda.
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That's a capital Greek lambda
there, the best I could do.
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So do you see that the two
equations written separately,
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or the four equations
or the n equations,
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combine into one
matrix equation.
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This is the same as
those two together.
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Good.
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But now that I have
it in matrix form,
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I can mess around with it.
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I can multiply both
sides by V inverse.
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If I multiply both sides by
V inverse I discover-- well,
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shall I multiply on
the left by V inverse?
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Yes, I'll do that.
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If I multiply on the left by
V inverse that's V inverse AV.
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This is matrix multiplication
and my next video
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is going to recap
matrix multiplication.
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So I multiply both
sides by V inverse.
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V inverse times V
is the identity.
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That's what the
inverse matrix is.
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V inverse, V is the identity.
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So there you go.
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Let me push that up.
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That's really nice.
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That's really nice.
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That's called diagonalizing
A. I diagonalize
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A by taking the eigenvector
matrix on the right,
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its inverse on the left,
multiply those three matrices,
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and I get this diagonal matrix.
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This is the diagonal
matrix lambda.
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Or other times I might want
to multiply by both sides
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here by V inverse
coming on the right.
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So that would give me A, V,
V inverse is the identity.
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So I can move V over
there as V inverse.
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That's what it amounts to.
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I multiply both
sides by V inverse.
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So this is just A and this is
the V, and the lambda, and now
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the V inverse.
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That's great.
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So that's a way to see how
A is built up or broken down
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into the eigenvector matrix,
times the eigenvalue matrix,
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times the inverse of
the eigenvector matrix.
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OK.
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Let me just use
that for a moment.
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Just so you see how it
connects with what we already
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know about eigenvalues
and eigenvectors.
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OK.
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So I'll copy that great fact,
that A is V lambda, V inverse.
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Oh, what do I want to do?
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I want to look at A squared.
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So if I look at
A squared, that's
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V lambda V inverse
times another one.
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Right?
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There's an A, there's an
A. So that's A squared.
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Well, you may say I've made
a mess out of A squared,
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but not true.
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V inverse V is the identity.
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So that it's just the identity
sitting in the middle.
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So the V at the far left,
then I have the lambda,
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and then I have the other
lambda-- lambda squared--
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and then the V inverse
at the far right.
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That's A squared.
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And if I did it n
times, I would have
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A to the n-th what would be
the lambda to the n-th power V
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inverse.
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What is this?
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What is this saying about?
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This is A squared.
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How do I understand
that equation?
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To me that says that the
eigenvalues of A squared
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are lambda squared.
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I'm just squaring
each eigenvalue.
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And the eigenvectors?
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What are the eigenvectors
of A squared?
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They're the same V, the
same vectors, x1, x2,
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that went into v. They're
also the eigenvectors
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of A squared, of A cubed, of
A to the n-th, of A inverse.
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So that's the point of
diagonalizing a matrix?
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Diagonalizing a
matrix is another way
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to see that when I square
the matrix, which is usually
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a big mess, looking at the
eigenvalues and eigenvectors
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it's the opposite of a big mess.
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It's very clear.
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The eigenvectors are
the same as for A.
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And the eigenvalues are squares
of the eigenvalues of A.
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In other words, we can
take the n-th power
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and we have a nice
notation for it.
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We learned already
that the n-th power
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has the eigenvalues
to the n-th power,
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and the eigenvectors the same.
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But now I just see it here.
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And there it is
for the n-th power.
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So if I took the same
matrix step 1,000 times,
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what would be important?
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What controls the thousandth
power of a matrix?
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The eigenvectors stay.
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They're just set.
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It would be the thousandth
power of the eigenvalue.
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So if this is a matrix with
an eigenvalue larger than 1,
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then the thousandth
power is going
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to be much larger than one.
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If this is a matrix with
eigenvalues smaller than 1,
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there are going to
be very small when
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I take the thousandth power.
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If there's an eigenvalue
that's exactly 1,
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that will be a steady state.
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And 1 to the thousandth
power will still be 1
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and nothing will change.
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So, the stability.
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What happens as I multiply,
take powers of a matrix,
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is a basic question parallel
to the question what
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happens with a
differential equation
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when I solve forward in time?
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I think of those two
problems as quite parallel.
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This is taking steps, single
steps, discrete steps.
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The differential equation is
moving forward continuously.
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This is a difference
between hop,
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hop, hop in the discrete case
and run forward continuously
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in the differential case.
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In both cases, the eigenvectors
and the eigenvalues
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are the guide to what
happens as time goes forward.
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OK.
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I have to do more about
working with matrices.
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Let me come to that next.
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Thanks.