WEBVTT
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GILBERT STRANG: OK.
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This is positive
definite matrix day.
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Our application was the
second-order equation
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with a symmetric matrix, S.
And we solved this equation.
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Second derivative, plus
S times y, equals 0.
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And you maybe remember
how we solved it.
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We looked for an
exponential solution.
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e to the I omega t,
times a vector x.
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We substituted,
and we discovered
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x had to be an eigenvector
of S, as usual.
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And lambda, which was omega
squared, is the eigenvalue.
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But I didn't stop to
point out that if we
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want lambda to be
omega squared, we
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need to know lambda
greater or equal to 0.
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So that is the best
of the best matrices.
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Symmetric and positive definite,
or positive semidefinite,
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which means the eigenvalues
are not only real,
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they're real for
symmetric matrices.
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They're also positive.
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Positive definite matrices--
automatically symmetric,
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I'm only talking about
symmetric matrices--
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and positive eigenvalues.
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OK.
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There it is.
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Positive definite matrix.
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All the eigenvalues
are positive.
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Positive semidefinite.
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That word semi allows
lambda equal 0.
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The matrix could be singular,
but all the eigenvalues
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have to be greater
or equal to 0.
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And let me show you exactly
where those matrices come from.
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Those matrices come
from A transpose A.
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If I take any matrix A,
could be rectangular.
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And A transpose
A will be square.
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A transpose A will be symmetric.
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And it will be at least
positive semidefinite.
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Why is that?
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This is the simple step
that is worth remembering.
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What's special about A
transpose A x equal lambda x?
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The good idea?
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Multiply both sides
by x transpose.
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Take the inner product
of both sides with x.
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Then I have x transpose
times the left side,
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is x transpose times
the right side.
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I'm OK with equation 2.
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When my S is A transpose
A, that's my S. OK.
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But now I look at this.
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That is A x in a
product with itself.
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That's the length
of A x squared.
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Any time I have y
transpose y, I'm
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getting the length of y squared.
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Here y is A x, so I'm getting
the length of A x squared.
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Over here, y is
x, so I'm getting
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the length of x squared.
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And you see that number
lambda is, in this equation,
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I have a number that
can't be negative.
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A number that's
positive, for sure.
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Because x is not the 0 vector.
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So lambda is never negative.
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A x could be the 0 vector.
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If we were in a singular case,
A x could be the 0 vector.
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In that case, I would only
learn lambda equals 0,
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and I'd be in this
semidefinite case.
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So if I want to move from
semidefinite to definite,
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then I must rule out
A x equals 0 there.
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Because that's
certainly a possibility.
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If I took the 0
matrix, all 0's, as A,
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A transpose A would
be the 0 matrix.
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That would be symmetric.
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All its eigenvalues would be 0.
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Would it be positive
semidefinite?
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Yes.
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Yes.
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All its eigenvalues
would actually be 0.
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Of course, that's not a
case that we are really
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thinking about.
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More often we're
in this good case
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where all the
eigenvalues are above 0.
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OK.
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So that's the meaning.
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And now the next job.
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How do we recognize a
positive definite matrix?
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It has to be symmetric.
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That's easy to see.
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But how can we tell if its
eigenvalues are positive?
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That's not fun because computing
eigenvalues is a big job.
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For a large matrix,
we take time on that.
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We didn't know how to do
it a little while ago.
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Now there are good
ways to do it,
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but it's not for paper
and pencil, and not for I.
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So how can we tell that all
the eigenvalues are positive?
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Well, we only want
to know their sign.
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We don't have to
know what they are.
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We don't know that we need the
number, we just want to know
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is it a positive number.
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And there are
several neat tests.
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Can I show you them?
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I'm going to have five tests.
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Five equivalent tests.
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Any one of these tests is
sufficient to make the matrix S
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positive definite.
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There is a particular S there
that I'll use as a test matrix.
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So there is a
symmetric matrix S.
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And I know it is
positive definite.
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But how do I know?
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OK.
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Well.
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So can you take
five things here?
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They connect all
of linear algebra.
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It's really beautiful.
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That the eigenvalues, that's
one chapter of linear algebra.
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The pivots are another
chapter of linear algebra.
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Do you remember about pivots?
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That's when you do elimination.
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So 4 is the first pivot.
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The first pivot.
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Pivot number 1 is the 4.
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And then when I take a multiple
of that away from that,
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I get a second pivot.
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And I'd see that
that was positive.
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So what's that?
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Maybe I take 1 and
1/2 away of this.
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I multiply that by
1 and 1/2, 6, 9.
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Subtract from 6, 10.
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So I actually get
a 1 down there.
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So pivot number 2
is a 1 in that case.
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Right?
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6, 9 taken away from
6, 10 leaves me 0, 1.
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OK.
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It passed the pivot test.
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Notice I didn't compute
the eigenvalues.
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I'm just doing other tests.
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Now here's another
beautiful test.
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It involves determinants.
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Now, I have to say upper.
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Upper left.
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Upper left determinants
greater than 0.
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What do I mean by an
upper left determinant?
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I look at my matrix.
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That's a 1 by 1 determinant.
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Certainly positive.
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That determinant is 4.
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Here is a 2 by 2 determinant.
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And that determinant is 40 minus
36, so happened to be 4 again.
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So the determinant
of the matrix is 4.
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But I also needed
the ones on the way.
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I can't just find
the determinant
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of the whole matrix.
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That's the last
part of this test,
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but I have to do all the
others as I get there.
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So it passes that test.
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Check.
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It works.
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So that test is passed.
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I'm doing more work than I need
to do because one test would
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have done the job.
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Now here comes another one.
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S is A transpose A. That's
what we looked at a minute ago.
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If S has this form
A transpose A.
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Oh, what did we
convince ourselves?
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We said that this was
sure to be semidefinite.
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And I needed some condition
to avoid A x equals 0.
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There was the possibility
of A x equals 0.
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I'll just bring that down.
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You remember we started
there and ended up here.
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And if A x was 0
then lambda was 0.
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We were in the
semidefinite case.
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So I have to avoid that.
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So I have to say when A
has independent columns.
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And I think I could factor that
matrix S into A transpose A.
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I'm sure I could.
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And get independent columns.
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And it would pass test 4.
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I want to go on to test 5.
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Which really, in a way, is
the definition, the best
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definition, of
positive definite.
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So if I took number 5,
it's the energy definition.
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So can I tell you
what that means?
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I mean that x transpose Sx.
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If I take my matrix S that I'm
testing for positive definite,
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I multiply on the
right by any vector
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x, any x, and on the
left by x transpose.
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Well, I get a number.
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S is a matrix.
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Sx is a vector.
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Inner product with a vector.
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I get a number.
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And that number should
be positive for all x.
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Oh, I have to make
one exception.
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If x is the 0 vector, then
of course that answer is 0.
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All x except the 0 vector.
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OK.
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So that would be a
way to-- another test.
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And this is associated in
applications with energy.
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So I call this the
energy test, or really
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the energy definition,
of positive definite.
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x transpose Sx.
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I'd like to apply that test.
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So you'll see what does it mean.
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Now we're looking at all x
to this particular example.
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But I won't throw
away this board here.
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You see eigenvalues, pivots,
determinants, A transpose A,
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and energy.
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Really all the pieces
of linear algebra.
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A transpose A. We'll
see it more and more.
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It comes up in least squares.
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If I have a general matrix
A, it's not even square.
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It doesn't have
great properties.
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But when I compute
A transpose A,
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then I get a symmetric matrix.
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And now I know also a
positive semidefinite.
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And with a little bit more
positive definite matrix.
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OK.
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By the way, are there five
tests for semidefinite matrices?
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Yes.
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There are five similar tests.
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All eigenvalues
greater or equal to 0.
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All pivots greater
or equal to 0.
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I can go down this and just
allow that borderline case
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that brings in semidefinite.
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I won't do that.
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Let me take my matrix S.
That small, example matrix.
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And apply the energy test.
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OK.
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So I'm looking at energy.
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So I'm looking at x.
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That's x1 x2, times my matrix
4, 6, 6, 10, times x1 x2.
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That's the energy.
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That's my x transpose Sx.
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x transpose Sx.
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Is that what we
wanted to compute?
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Yes.
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x transpose Sx.
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Now, can I compute that?
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Yes.
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It's a matrix multiplication.
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Nothing magical here.
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But when I do, I'll
show you the shortcut.
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When I do that, a 4
x1 is going to appear,
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and it'll be multiplied
by that x1 over there.
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I'll get a 4 x1 squared.
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And then I'll have a 6 x2
that's multiplying that x1.
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So there's a 6 x1 x2.
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And now from this.
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That was the first component.
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And now I have 6 x1 and 10 x2.
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Multiply an x2.
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Well, that's another 6 x1 x2.
00:13:19.830 --> 00:13:24.230
And the 10, we'll
multiply x2 and x2.
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x2 squared.
00:13:26.290 --> 00:13:28.390
I did that quickly.
00:13:28.390 --> 00:13:32.950
But the result is
just easy to see.
00:13:32.950 --> 00:13:37.300
The 4, 6, 6, 10 show
up in the squares.
00:13:37.300 --> 00:13:40.710
The diagonal 4 and 10
show up in the squares.
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And the off diagonal
6, which doubles to 12,
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shows up in the x1
x2, the cross term.
00:13:50.860 --> 00:13:51.650
OK.
00:13:51.650 --> 00:13:57.410
Now why should that-- so that's
a number for every x1 and x2.
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Suppose x1 is 1 and x2 is 1.
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Then the number I get is
4, plus 6, plus 6, plus 10.
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That's probably 26.
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It's positive.
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What if x1 is 1--
let me try this.
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x1 is 1 and x2 is minus 1.
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Do I still get a
positive energy?
00:14:20.340 --> 00:14:23.550
So my vector is 1 minus 1.
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So I get 4.
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Now, because of that, I
have minus 6, and minus 6,
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and 10, from the x2 squared.
00:14:34.790 --> 00:14:36.710
And that's 14 minus 12.
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That's 2.
00:14:38.197 --> 00:14:38.780
It's positive.
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Well, I tested two vectors.
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I tested the 1, 1 vector
and the 1, minus 1 vector.
00:14:48.800 --> 00:14:53.300
But you have to know
that for every vector x,
00:14:53.300 --> 00:14:55.660
this does turn out
to be positive.
00:14:55.660 --> 00:14:57.910
And I can show you
that by something
00:14:57.910 --> 00:15:01.640
called completing the square.
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It's not what I plan to do.
00:15:04.520 --> 00:15:09.600
But the beauty is we now
understand this energy test.
00:15:09.600 --> 00:15:15.230
What it means to take x
transpose Sx, write it out,
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and ask is it always positive.
00:15:18.520 --> 00:15:20.290
Is it always positive?
00:15:20.290 --> 00:15:20.790
OK.
00:15:20.790 --> 00:15:24.250
So that's the fifth,
number 5, test.
00:15:24.250 --> 00:15:27.250
But I think of it really
as the definition.
00:15:27.250 --> 00:15:30.530
And it means-- can
I draw a picture?
00:15:30.530 --> 00:15:32.050
Here is x1.
00:15:32.050 --> 00:15:33.410
Here's x2.
00:15:33.410 --> 00:15:36.470
And now I'm going to--
this is my function.
00:15:39.070 --> 00:15:40.490
x transpose A x.
00:15:40.490 --> 00:15:42.310
My energy.
00:15:42.310 --> 00:15:49.110
If I graph that, I have an
x, and a y, and a function z.
00:15:49.110 --> 00:15:51.320
That function of x and y.
00:15:51.320 --> 00:15:53.450
What kind of a
graph does it have?
00:15:53.450 --> 00:15:57.850
When x1 and x2
are 0, it's there.
00:15:57.850 --> 00:16:02.420
When x1 and x2 move away
from 0, it goes positive.
00:16:02.420 --> 00:16:05.630
That graph is like that.
00:16:05.630 --> 00:16:06.670
It's sort of a bowl.
00:16:10.060 --> 00:16:12.180
And I have a minimum.
00:16:16.370 --> 00:16:20.850
One of the main application
of derivatives in calculus
00:16:20.850 --> 00:16:24.760
is to find the
test for a minimum,
00:16:24.760 --> 00:16:27.810
and decide minimum or maximum.
00:16:27.810 --> 00:16:29.570
Minimum or maximum.
00:16:29.570 --> 00:16:31.410
And you remember the
second derivative
00:16:31.410 --> 00:16:33.780
decides a minimum or maximum.
00:16:33.780 --> 00:16:36.570
Positive second
derivative, minimum.
00:16:36.570 --> 00:16:38.840
Negative second
derivative, maximum.
00:16:38.840 --> 00:16:41.840
It tells you about the
bending of the curve.
00:16:41.840 --> 00:16:46.840
Well, we're in two
dimensions now,
00:16:46.840 --> 00:16:50.090
with a function
of two variables.
00:16:50.090 --> 00:16:52.580
This is multivariable calculus.
00:16:52.580 --> 00:16:57.030
So what becomes positive
second derivative,
00:16:57.030 --> 00:17:00.190
becomes positive
definite matrix.
00:17:00.190 --> 00:17:03.306
A matrix of second derivatives.
00:17:03.306 --> 00:17:06.950
This is the whole
subject of optimization.
00:17:06.950 --> 00:17:09.839
Maximizing,
minimizing, comes here.
00:17:09.839 --> 00:17:10.339
OK.
00:17:10.339 --> 00:17:13.140
That's for another day.
00:17:13.140 --> 00:17:14.819
I just would like
to tell you one more
00:17:14.819 --> 00:17:17.490
thing about positive
definite matrices.
00:17:17.490 --> 00:17:21.530
I got a book in the mail
which could be quite valuable.
00:17:21.530 --> 00:17:24.420
It's a little
paperback, and the title
00:17:24.420 --> 00:17:35.140
is Frequently Asked Questions in
Interviews for Financial Math.
00:17:35.140 --> 00:17:36.460
Being a Quant.
00:17:36.460 --> 00:17:37.560
Going to Wall Street.
00:17:37.560 --> 00:17:39.610
Becoming rich.
00:17:39.610 --> 00:17:43.580
So they don't give you
all the money right away.
00:17:43.580 --> 00:17:47.770
They make you show that
you know something.
00:17:47.770 --> 00:17:52.120
And so they ask a
few math questions.
00:17:52.120 --> 00:17:57.630
And the first question was--
I was happy to see this.
00:17:57.630 --> 00:18:04.390
The first question asked,
when is this matrix
00:18:04.390 --> 00:18:06.670
positive definite?
00:18:10.860 --> 00:18:11.360
OK.
00:18:11.360 --> 00:18:12.560
Can you see that matrix?
00:18:12.560 --> 00:18:14.370
1 is on the diagonal.
00:18:14.370 --> 00:18:15.910
Those are correlation.
00:18:15.910 --> 00:18:17.640
This is a correlation matrix.
00:18:17.640 --> 00:18:19.780
That's why it's
important in finance.
00:18:19.780 --> 00:18:21.930
It might be the
three correlations
00:18:21.930 --> 00:18:28.140
of bonds, and stocks,
and foreign exchange.
00:18:28.140 --> 00:18:32.350
So each one is
correlated to itself
00:18:32.350 --> 00:18:35.590
with a full correlation of 1.
00:18:35.590 --> 00:18:38.470
But there'll be a correlation
between bonds and stocks
00:18:38.470 --> 00:18:40.950
going up together,
but not perfectly
00:18:40.950 --> 00:18:43.770
together, by some number a.
00:18:43.770 --> 00:18:47.810
And bonds and foreign
exchange with some number b.
00:18:47.810 --> 00:18:52.630
Stocks and foreign
exchange, some number c.
00:18:52.630 --> 00:18:55.510
So that's the matrix
of correlations.
00:18:55.510 --> 00:18:59.810
And the key point is,
it is positive definite.
00:18:59.810 --> 00:19:03.520
So the question when
you go to Wall Street
00:19:03.520 --> 00:19:05.570
to apply for the money.
00:19:05.570 --> 00:19:12.130
If you're asked what's the test
on those numbers a, b, c, to
00:19:12.130 --> 00:19:18.060
have a positive definite,
proper correlation matrix?
00:19:18.060 --> 00:19:20.920
I would suggest the
determinant test.
00:19:20.920 --> 00:19:24.180
The determinant test, if
I'm given a small matrix,
00:19:24.180 --> 00:19:25.950
I'll just do the determinants.
00:19:25.950 --> 00:19:27.690
So that determinant is 1.
00:19:27.690 --> 00:19:29.030
No problem.
00:19:29.030 --> 00:19:32.600
This determinant, what's
the 2 by 2 determinant?
00:19:32.600 --> 00:19:34.530
1 minus a squared.
00:19:34.530 --> 00:19:38.330
So 1 minus a squared
has to be positive.
00:19:38.330 --> 00:19:40.450
I'm doing the determinant test.
00:19:40.450 --> 00:19:43.260
And what's the 3
by 3 determinant?
00:19:43.260 --> 00:19:45.930
1 from the diagonal.
00:19:45.930 --> 00:19:49.770
And I have an acb and an acb.
00:19:49.770 --> 00:19:57.250
I think I have two acb's
from the three terms.
00:19:57.250 --> 00:20:02.230
Now, those terms
have the plus signs.
00:20:02.230 --> 00:20:05.020
And now I have some
with a minus sign,
00:20:05.020 --> 00:20:06.850
which better not be too big.
00:20:06.850 --> 00:20:10.290
That's the whole point on
positive definite matrices.
00:20:10.290 --> 00:20:15.450
The off diagonal is not allowed
to overrun the diagonal.
00:20:15.450 --> 00:20:19.550
The diagonal should be
the biggest numbers.
00:20:19.550 --> 00:20:20.140
OK.
00:20:20.140 --> 00:20:25.280
So I saw that a squared
had to be below 1.
00:20:25.280 --> 00:20:27.730
But now what's the
determinant test?
00:20:27.730 --> 00:20:30.910
I think this has to
be bigger than what
00:20:30.910 --> 00:20:33.090
I'm getting from
this direction, which
00:20:33.090 --> 00:20:36.260
is a b squared, and a c
squared, and an a squared.
00:20:36.260 --> 00:20:37.410
Oh, look at that.
00:20:37.410 --> 00:20:40.530
a squared, b squared,
and c squared.
00:20:43.260 --> 00:20:45.810
That would be the answer.
00:20:45.810 --> 00:20:48.700
That first test there.
00:20:48.700 --> 00:20:51.340
Second test there.
00:20:51.340 --> 00:20:55.170
Well, the easy test was
just 1, is positive.
00:20:55.170 --> 00:20:59.250
So really, that's what
they're looking for.
00:20:59.250 --> 00:21:01.710
That would be the
test, those numbers.
00:21:01.710 --> 00:21:08.130
So abc can't be too large
or that would begin to fail.
00:21:08.130 --> 00:21:09.180
Good.
00:21:09.180 --> 00:21:14.260
So positive definite matrices
have lots of applications.
00:21:14.260 --> 00:21:15.870
Here was minimum.
00:21:15.870 --> 00:21:19.560
Here was correlation
matrices and finance.
00:21:19.560 --> 00:21:21.380
Many, many other places.
00:21:21.380 --> 00:21:28.160
Let me just bring
down the five tests.
00:21:28.160 --> 00:21:31.250
Eigenvalues, pivots,
determinants, A transpose A,
00:21:31.250 --> 00:21:33.290
and energy.
00:21:33.290 --> 00:21:34.670
And I'll stop there.
00:21:34.670 --> 00:21:36.340
Thank you.