WEBVTT
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-- one and --
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the lecture on
symmetric matrixes.
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So that's the most
important class
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of matrixes, symmetric matrixes.
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A equals A transpose.
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So the first points, the
main points of the lecture
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I'll tell you right away.
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What's special about
the eigenvalues?
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What's special about
the eigenvectors?
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This is -- the way we
now look at a matrix.
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We want to know about its
eigenvalues and eigenvectors
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and if we have a
special type of matrix,
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that should tell us
something about eigenvalues
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and eigenvectors.
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Like Markov matrixes, they
have an eigenvalue equal
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one.
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Now symmetric matrixes, can I
just tell you right off what
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the main facts -- the
two main facts are?
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The eigenvalues of a
symmetric matrix, real --
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this is a real
symmetric matrix, we --
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talking mostly
about real matrixes.
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The eigenvalues are also real.
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So our examples of
rotation matrixes, where --
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where we got E- eigenvalues
that were complex,
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that won't happen now.
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For symmetric matrixes,
the eigenvalues are real
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and the eigenvectors
are also very special.
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The eigenvectors are
perpendicular, orthogonal,
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so which do you prefer?
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I'll say perpendicular.
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Perp- well, they're
both long words.
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Okay, right.
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So -- I have a --
you should say "why?"
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and I'll at least answer why
for case one, maybe case two,
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the checking the Eigen --
that the eigenvectors are
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perpendicular, I'll leave
to, the -- to the book.
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But let's just realize what --
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well, first I have to say, it --
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it could happen, like for
the identity matrix --
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there's a symmetric matrix.
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Its eigenvalues are
certainly all real,
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they're all one for
the identity matrix.
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What about the eigenvectors?
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Well, for the identity, every
vector is an eigenvector.
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So how can I say
they're perpendicular?
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What I really mean
is the -- they --
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this word are should really
be written can be chosen
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perpendicular.
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That is, if we have --
it's the usual case.
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If the eigenvalues
are all different,
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then each eigenvalue has
one line of eigenvectors
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and those lines are
perpendicular here.
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But if an eigenvalue's
repeated, then there's
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a whole plane of eigenvectors
and all I'm saying
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is that in that plain, we can
choose perpendicular ones.
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So that's why it's a can
be chosen part, is --
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this is in the case of a
repeated eigenvalue where
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there's some real,
substantial freedom.
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But the typical case is
different eigenvalues,
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all real, one dimensional
eigenvector space,
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Eigen spaces, and
all perpendicular.
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So, just -- let's just
see the conclusion.
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If we accept those as
correct, what happens --
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and I also mean that
there's a full set of them.
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so forgive me for doing such
a thing, but, I'll look at the
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I -- so that's part of this
picture here, that there --
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there's a complete
set of eigenvectors,
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perpendicular ones.
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So, having a complete set
of eigenvectors means --
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so normal --
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so the usual -- maybe I put
the -- usually -- usual --
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usual case is that the matrix
A we can write in terms of its
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eigenvalue matrix and its
eigenvector matrix this way,
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right?
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We can do that in
the usual case,
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but now what's special when
the matrix is symmetric?
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So this is the
usual case, and now
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let me go to the symmetric case.
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So in the symmetric
case, A, this --
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this should become
somehow a little special.
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Well, the lambdas on the
diagonal are still on the
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diagonal.
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They're -- they're real,
but that's where they are.
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What about the
eigenvector matrix?
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So what can I do now special
about the eigenvector matrix
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when -- when the A
itself is symmetric,
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that says something good
about the eigenvector matrix,
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so what is this --
what does this lead to?
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This -- these perpendicular
eigenvectors, I can not only --
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I can not only guarantee
they're perpendicular,
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I could also make them
unit vectors, no problem,
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just s- scale their length to
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one.
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So what do I have?
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I have orthonormal eigenvectors.
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And what does that tell me
about the eigenvector matrix?
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What -- what letter should
I now use in place of S --
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I've got -- those two equations
are identical,1 remember S has
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the eigenvectors in its columns,
but now those columns are
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orthonormal, so the
right letter to use is Q.
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So that's where -- so we've got
the letter all set up, book.
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so this should be
Q lambda Q inverse.
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Q standing in our minds always
for this matrix -- in this case
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it's square, it's --
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so these are the Okay.
columns of Q, of course.
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And one more thing.
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What's Q inverse?
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For a matrix that has
these orthonormal columns,
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So I took the dot product
-- ye, somehow, it didn't --
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I we know that the inverse
is the same as the transpose.
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So here is the beautiful --
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there is the -- the great
haven't learned anything.
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description, the factorization
of a symmetric matrix.
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And this is, like, one
of the famous theorems
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of linear algebra, that if
I have a symmetric matrix,
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it can be factored in this form.
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An orthogonal matrix
times diagonal times
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the transpose of that
orthogonal matrix.
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And, of course, everybody
immediately says yes,
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and if this is
possible, then that's
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clearly symmetric, right?
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That -- take -- we've looked
at products of three guys like
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that and taken their transpose
and we got it back again.
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So do you -- do you see
the beauty of this --
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of this factorization, then?
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It -- it completely displays
the eigenvalues and eigenvectors
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the symmetry of the -- of
the whole thing, because --
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that product, Q times
lambda times Q transpose,
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if I transpose it, it -- this
comes in this position and we
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get that matrix back again.
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So that's -- in mathematics,
that's called the spectral
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Spectrum is the set of
eigenvalues of a matrix.
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theorem.
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Spec- it somehow comes from the
idea of the spectrum of light
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as a combination
of pure things --
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where our matrix is broken
down into pure eigenvalues
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and eigenvectors --
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in mechanics it's often called
the principle axis theorem.
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It's very useful.
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It means that if you have --
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we'll see it geometrically.
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It means that if I
have some material --
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if I look at the right axis, it
becomes diagonal, it becomes --
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the -- the
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I- I've done something dumb,
because I've got the --
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I should've taken the dot
product of this guy here with
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-- that's directions
don't couple together.
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Okay.
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So that's -- that -- that's
what to remember from --
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from this lecture.
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Now, I would like to say why
are the eigenvalues real?
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Can I do that?
00:09:07.080 --> 00:09:10.200
So -- so -- because that --
something useful comes out.
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So I'll just come back --
come to that question why real
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eigenvalues?
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Okay.
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So I have to start
from the only thing
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we know, Ax equal lambda x.
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Okay.
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But as far as I
know at this moment,
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lambda could be complex.
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I'm going to prove it's not
-- and x could be complex.
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In fact, for the moment,
even A could be --
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we could even think, well,
what happens if A is complex?
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Well, one thing we can
always do -- this is --
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this is like always --
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always okay --
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I can -- if I have an equation,
I can take the complex
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conjugate of everything.
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That's -- no -- no -- so A
conjugate x conjugate equal
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lambda conjugate x conjugate, it
just means that everywhere over
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here that there was a -- an
equals x bar transpose lambda
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bar x bar. i, then here
I changed it to a-i.
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That's -- that -- you
know that that step --
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that conjugate
business, that a+ib,
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if I conjugate it it's a-ib.
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That's the meaning of conjugate
-- and products behave right,
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I can conjugate every factor.
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So I haven't done anything yet
except to say what would be
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true if, x -- in any case, even
if x and lambda were complex.
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Of course, our -- we're
speaking about real matrixes A,
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so I can take that out.
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Actually, this already tells me
something about real matrixes.
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I haven't used any
assumption of A --
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A transpose yet.
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Symmetry is waiting in
the wings to be used.
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This tells me that if a real
matrix has an eigenvalue lambda
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what I was going to do. and an
eigenvector x, it also has --
00:11:22.040 --> 00:11:25.040
another of its
eigenvalues is lambda bar
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with eigenvector x bar.
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Real matrixes, the eigenvalues
come in lambda, lambda bar --
00:11:32.500 --> 00:11:37.500
the complex eigenvalues come
in lambda and lambda bar pairs.
00:11:37.500 --> 00:11:39.600
But, of course,
I'm aiming to show
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that they're not complex at all,
here, by getting symmetry in.
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So how I going to use symmetry?
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I'm going to transpose
this equation to x bar
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transpose A transpose equals
x bar transpose lambda bar.
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That's just a number, so I don't
mind wear I put that number.
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This is -- this is --
00:12:05.750 --> 00:12:06.480
this is a -- then
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again okay.
00:12:08.250 --> 00:12:10.000
Ax equals lambda x bar
transpose x, right?
00:12:10.000 --> 00:12:14.880
But now I'm ready
to use symmetry.
00:12:14.880 --> 00:12:18.890
I'm ready -- so this
was all just mechanics.
00:12:18.890 --> 00:12:22.040
Now -- now comes the
moment to say, okay,
00:12:22.040 --> 00:12:24.040
if the matrix is this
from the right with x bar,
00:12:24.040 --> 00:12:25.665
I get x bar transpose
Ax bar symmetric,
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then this A transpose
is the same as A.
00:12:29.030 --> 00:12:31.910
You see, at that moment
I used the assumption.
00:12:31.910 --> 00:12:34.380
Now let me finish
the discussion.
00:12:34.380 --> 00:12:37.610
Here -- here's the way I finish.
00:12:37.610 --> 00:12:42.460
I look at this original equation
and I take the inner product.
00:12:42.460 --> 00:12:45.170
I multiply both sides by --
00:12:45.170 --> 00:12:46.680
oh, maybe I'll do
it with this one.
00:12:49.260 --> 00:12:51.540
I take --
00:12:51.540 --> 00:12:55.200
I multiply both sides
by x bar transpose.
00:12:55.200 --> 00:12:58.670
x bar transpose Ax
bar equals lambda
00:12:58.670 --> 00:13:03.620
bar x bar transpose x bar.
00:13:03.620 --> 00:13:04.470
Okay, fine.
00:13:08.330 --> 00:13:12.170
All right, now
what's the other one?
00:13:12.170 --> 00:13:16.880
Oh, for the other one I'll
probably use this guy.
00:13:16.880 --> 00:13:20.580
A- I happy about this?
00:13:20.580 --> 00:13:21.080
No.
00:13:21.080 --> 00:13:22.910
For some reason I'm not.
00:13:22.910 --> 00:13:26.830
I'm -- I want to --
00:13:26.830 --> 00:14:04.850
if I take the inner
product of Okay.
00:14:24.690 --> 00:14:26.660
So that -- that
was -- that's fine.
00:14:26.660 --> 00:14:30.230
That comes directly from that,
multiplying both sides by x bar
00:14:30.230 --> 00:14:33.860
transpose, but now
I'd like to get --
00:14:33.860 --> 00:14:38.320
why do I have x bars over there?
00:14:38.320 --> 00:14:40.860
Oh, yes.
00:14:40.860 --> 00:14:43.161
Forget this.
00:14:43.161 --> 00:14:43.660
Okay.
00:14:43.660 --> 00:14:45.820
On this one -- right.
00:14:45.820 --> 00:14:49.010
On this one, I took it like
that, I multiply on the right
00:14:49.010 --> 00:14:50.400
by x.
00:14:50.400 --> 00:14:54.080
That's the idea.
00:14:54.080 --> 00:14:55.540
Okay.
00:14:55.540 --> 00:15:02.150
Now why I happier with
this situation now?
00:15:02.150 --> 00:15:04.180
A proof is coming here.
00:15:04.180 --> 00:15:10.140
Because I compare this
guy with this one.
00:15:10.140 --> 00:15:12.570
And they have the
same left hand side.
00:15:12.570 --> 00:15:14.410
So they have the
same right hand side.
00:15:14.410 --> 00:15:16.240
So comparing those two, can --
00:15:16.240 --> 00:15:19.910
I'll raise the board to
do this comparison --
00:15:19.910 --> 00:15:25.150
this thing, lambda
x bar transpose x
00:15:25.150 --> 00:15:32.880
is equal to lambda
bar x bar transpose x.
00:15:32.880 --> 00:15:33.380
Okay.
00:15:35.890 --> 00:15:38.190
And the conclusion
I'm going to reach --
00:15:38.190 --> 00:15:43.280
I -- I on the right track here?
00:15:43.280 --> 00:15:44.830
The conclusion
I'm going to reach
00:15:44.830 --> 00:15:47.060
is lambda equal lambda bar.
00:15:52.170 --> 00:15:55.330
I would have to track down the
other possibility that this --
00:15:55.330 --> 00:15:58.400
this thing is
zero, but let me --
00:15:58.400 --> 00:16:01.330
oh -- oh, yes, that's important.
00:16:01.330 --> 00:16:03.370
It's not zero.
00:16:03.370 --> 00:16:08.760
So once I know that this
isn't zero, I just cancel it
00:16:08.760 --> 00:16:11.220
and I learn that lambda
equals lambda bar.
00:16:11.220 --> 00:16:14.360
And so what can you -- do you --
00:16:14.360 --> 00:16:17.823
have you got the
reasoning altogether?
00:16:20.640 --> 00:16:24.130
What does this tell us?
00:16:24.130 --> 00:16:27.750
Lambda's an eigenvalue
of this symmetric matrix.
00:16:27.750 --> 00:16:30.300
We've just proved that
it equaled lambda bar,
00:16:30.300 --> 00:16:34.630
so we have just proved
that lambda is real,
00:16:34.630 --> 00:16:35.710
right?
00:16:35.710 --> 00:16:39.980
If, if a number is equal to
its own complex conjugate,
00:16:39.980 --> 00:16:42.290
then there's no
imaginary part at all.
00:16:42.290 --> 00:16:43.280
The number is real.
00:16:43.280 --> 00:16:45.435
So lambda is real.
00:16:48.260 --> 00:16:49.380
Good.
00:16:49.380 --> 00:16:51.740
Good.
00:16:51.740 --> 00:16:56.310
Now, what -- but it depended
on this little expression,
00:16:56.310 --> 00:16:59.040
on knowing that
that wasn't zero,
00:16:59.040 --> 00:17:03.960
so that I could cancel it out
-- so can we just take a second
00:17:03.960 --> 00:17:05.400
on that one?
00:17:05.400 --> 00:17:08.650
Because it's an
important quantity.
00:17:08.650 --> 00:17:11.160
x bar transpose x.
00:17:11.160 --> 00:17:18.280
Okay, now remember, as far
as we know, x is complex.
00:17:18.280 --> 00:17:20.650
So this is --
00:17:20.650 --> 00:17:25.260
here -- x is complex, x
has these components, x1,
00:17:25.260 --> 00:17:28.310
x2 down to xn.
00:17:28.310 --> 00:17:35.890
And x bar transpose, well, it's
transposed and it's conjugated,
00:17:35.890 --> 00:17:42.570
so that's x1 conjugated x2
conjugated up to xn conjugated.
00:17:42.570 --> 00:17:43.520
I'm -- I'm --
00:17:43.520 --> 00:17:46.390
I'm really reminding
you of crucial facts
00:17:46.390 --> 00:17:48.440
about complex numbers
that are going
00:17:48.440 --> 00:17:51.230
to come into the next
lecture as well as this one.
00:17:51.230 --> 00:17:57.700
So w- what can you tell
me about that product --
00:17:57.700 --> 00:18:01.150
I -- I guess what
I'm trying to say is,
00:18:01.150 --> 00:18:05.830
if I had a complex vector, this
would be the quantity I would
00:18:05.830 --> 00:18:06.330
--
00:18:06.330 --> 00:18:07.730
I would like.
00:18:07.730 --> 00:18:09.310
This is the quantity I like.
00:18:09.310 --> 00:18:13.390
I would take the vector times
its transpose -- now what --
00:18:13.390 --> 00:18:17.070
what happens usually if I take a
vector -- a -- a -- x transpose
00:18:17.070 --> 00:18:18.070
x?
00:18:18.070 --> 00:18:22.330
I mean, that's a quantity we
see all the time, x transpose x.
00:18:22.330 --> 00:18:25.200
That's the length
of x squared, right?
00:18:25.200 --> 00:18:28.350
That's this positive length
squared, it's Pythagoras,
00:18:28.350 --> 00:18:31.640
it's x1 squared plus
x2 squared and so on.
00:18:31.640 --> 00:18:36.100
Now our vector's complex,
and you see the effect?
00:18:36.100 --> 00:18:39.090
I'm conjugating
one of these guys.
00:18:39.090 --> 00:18:41.180
So now when I do
this multiplication,
00:18:41.180 --> 00:18:49.370
I have x1 bar times x1 and
x2 bar times x2 and so on.
00:18:49.370 --> 00:18:53.640
So this is an --
this is sum a+ib.
00:18:53.640 --> 00:18:57.960
And this is sum a-ib.
00:18:57.960 --> 00:19:02.440
I mean, what's the point here?
00:19:02.440 --> 00:19:05.450
What's the point -- when
I multiply a number by its
00:19:05.450 --> 00:19:11.480
conjugate, a complex number by
its conjugate, what do I get?
00:19:11.480 --> 00:19:16.140
I get a n- the -- the
imaginary part is gone.
00:19:16.140 --> 00:19:20.680
When I multiply a+ib by
its conjugate, what's --
00:19:20.680 --> 00:19:23.400
what's the result of that -- of
each of those separate little
00:19:23.400 --> 00:19:25.210
multiplications?
00:19:25.210 --> 00:19:29.970
There's an a squared and -- and
what -- how many -- what's --
00:19:29.970 --> 00:19:34.010
b squared comes in
with a plus or a minus?
00:19:34.010 --> 00:19:35.290
A plus.
00:19:35.290 --> 00:19:39.260
i times minus i is
a plus b squared.
00:19:39.260 --> 00:19:41.240
And what about the
imaginary part?
00:19:43.830 --> 00:19:46.770
Gone, right?
00:19:46.770 --> 00:19:49.050
An iab and a minus iab.
00:19:49.050 --> 00:19:52.870
So this -- this is
the right thing to do.
00:19:52.870 --> 00:20:00.600
If you want a decent answer,
then multiply numbers
00:20:00.600 --> 00:20:02.910
by their conjugates.
00:20:02.910 --> 00:20:08.900
Multiply vectors by the
conjugates of x transpose.
00:20:08.900 --> 00:20:13.750
So this quantity is positive,
this quantity is positive --
00:20:13.750 --> 00:20:17.460
the whole thing is positive
except for the zero vector
00:20:17.460 --> 00:20:23.010
and that allows me to know
that this is a positive number,
00:20:23.010 --> 00:20:27.220
which I safely cancel out
and I reach the conclusion.
00:20:27.220 --> 00:20:33.290
So actually, in this discussion
here, I've done two things.
00:20:33.290 --> 00:20:35.690
If I reached the
conclusion that lambda's
00:20:35.690 --> 00:20:38.510
real, which I wanted to do.
00:20:38.510 --> 00:20:41.600
But at the same time,
we sort of saw what
00:20:41.600 --> 00:20:43.790
to do if things were complex.
00:20:43.790 --> 00:20:48.490
If a vector is complex,
then it's x bar transpose x,
00:20:48.490 --> 00:20:55.350
this is its length squared.
00:20:55.350 --> 00:20:59.510
And as I said, the next
lecture Monday, we'll --
00:20:59.510 --> 00:21:03.220
we'll repeat that this is
the right thing and then do
00:21:03.220 --> 00:21:06.280
the right thing for
matrixes and all other --
00:21:06.280 --> 00:21:10.950
all other, complex
possibilities.
00:21:10.950 --> 00:21:12.070
Okay.
00:21:12.070 --> 00:21:17.070
But the main point, then,
is that the eigenvalues
00:21:17.070 --> 00:21:20.720
of a symmetric matrix, it
just -- do you -- do --
00:21:20.720 --> 00:21:23.620
where did we use
symmetry, by the way?
00:21:23.620 --> 00:21:24.860
We used it here, right?
00:21:24.860 --> 00:21:27.870
Let -- can I just --
00:21:27.870 --> 00:21:31.980
let -- suppose A was a complex.
00:21:31.980 --> 00:21:34.800
Suppose A had been
a complex number.
00:21:34.800 --> 00:21:37.230
Could -- could I have
made all this work?
00:21:37.230 --> 00:21:40.780
If A was a complex
number -- complex matrix,
00:21:40.780 --> 00:21:45.070
then here I should
have written A bar.
00:21:45.070 --> 00:21:47.600
I erased the bar because
I assumed A was real.
00:21:47.600 --> 00:21:50.560
But now let's suppose
for a moment it's not.
00:21:50.560 --> 00:21:55.260
Then when I took this
step, what should I have?
00:21:55.260 --> 00:21:56.710
What did I do on that step?
00:21:56.710 --> 00:21:57.820
I transposed.
00:21:57.820 --> 00:22:00.215
So I should have
A bar transpose.
00:22:03.560 --> 00:22:05.990
In the symmetric
case, that was A,
00:22:05.990 --> 00:22:08.590
and that's what made
everything work, right?
00:22:08.590 --> 00:22:12.720
This -- this led
immediately to that.
00:22:12.720 --> 00:22:17.840
This one led immediately to
this when the matrix was real,
00:22:17.840 --> 00:22:20.300
so that didn't matter,
and it was symmetric,
00:22:20.300 --> 00:22:21.790
so that didn't matter.
00:22:21.790 --> 00:22:23.600
Then I got A.
00:22:23.600 --> 00:22:26.760
But -- so now I
just get to ask you.
00:22:26.760 --> 00:22:30.900
Suppose the matrix
had been complex.
00:22:30.900 --> 00:22:35.620
What's the right equivalent
of sym- symmetry?
00:22:38.320 --> 00:22:41.480
So the good matrix --
so here, let me say --
00:22:41.480 --> 00:22:52.776
good matrixes -- by good I mean
real lambdas and perpendicular
00:22:52.776 --> 00:22:53.276
x-s.
00:22:57.310 --> 00:23:02.860
And tell me now, which
matrixes are good?
00:23:02.860 --> 00:23:04.800
If they're --
00:23:04.800 --> 00:23:08.070
If they're real
matrixes, the good ones
00:23:08.070 --> 00:23:10.790
are symmetric, because then
everything went through.
00:23:10.790 --> 00:23:13.010
The -- so the good --
00:23:13.010 --> 00:23:15.050
I'm saying now what's good.
00:23:15.050 --> 00:23:17.430
This is -- this is -- these
are the good matrixes.
00:23:17.430 --> 00:23:21.030
They have real eigenvalues,
perpendicular eigenvectors --
00:23:21.030 --> 00:23:27.910
good means A equal
A transpose if real.
00:23:27.910 --> 00:23:30.560
Then -- then that was
what -- our proof worked.
00:23:30.560 --> 00:23:35.690
But if A is complex, all -- our
proof will still work provided
00:23:35.690 --> 00:23:38.060
A bar transpose is A.
00:23:38.060 --> 00:23:41.740
Do you see what I'm saying?
00:23:41.740 --> 00:23:47.700
I'm saying if we have complex
matrixes and we want to say are
00:23:47.700 --> 00:23:51.330
they -- are they as good
as symmetric matrixes,
00:23:51.330 --> 00:23:56.750
then we should not only
transpose the thing,
00:23:56.750 --> 00:23:58.760
but conjugate it.
00:23:58.760 --> 00:24:00.800
Those are good matrixes.
00:24:00.800 --> 00:24:03.090
And of course,
the most important
00:24:03.090 --> 00:24:06.980
s- the most important
case is when they're real,
00:24:06.980 --> 00:24:09.410
this part doesn't
matter and I just have
00:24:09.410 --> 00:24:11.330
A equal A transpose symmetric.
00:24:11.330 --> 00:24:12.070
Do you -- I --
00:24:12.070 --> 00:24:15.260
I'll just repeat that.
00:24:15.260 --> 00:24:20.290
The good matrixes, if
complex, are these.
00:24:20.290 --> 00:24:23.590
If real, that doesn't
make any difference
00:24:23.590 --> 00:24:25.900
so I'm just saying symmetric.
00:24:25.900 --> 00:24:30.530
And of course, 99% of
examples and applications
00:24:30.530 --> 00:24:34.740
to the matrixes are real
and we don't have that
00:24:34.740 --> 00:24:38.230
and then symmetric
is the key property.
00:24:38.230 --> 00:24:40.950
Okay.
00:24:40.950 --> 00:24:48.170
So that -- that's, these main
facts and now let me just --
00:24:48.170 --> 00:24:53.690
let me just -- so that's
this x bar transpose x, okay.
00:24:53.690 --> 00:24:59.210
So I'll just, write it
once more in this form.
00:24:59.210 --> 00:25:05.370
So perpendicular orthonormal
eigenvectors, real eigenvalues,
00:25:05.370 --> 00:25:08.410
transposes of
orthonormal eigenvectors.
00:25:08.410 --> 00:25:13.690
That's the symmetric
case, A equal A transpose.
00:25:13.690 --> 00:25:15.320
Okay.
00:25:15.320 --> 00:25:18.030
Good.
00:25:18.030 --> 00:25:23.340
Actually, I'll even
take one more step here.
00:25:23.340 --> 00:25:25.860
Suppose -- I --
00:25:25.860 --> 00:25:29.350
I can break this down
to show you really
00:25:29.350 --> 00:25:34.000
what that says about
a symmetric matrix.
00:25:34.000 --> 00:25:35.180
I can break that down.
00:25:35.180 --> 00:25:40.110
Let me here -- here
go these eigenvectors.
00:25:40.110 --> 00:25:46.540
I -- here go these eigenvalues,
lambda one, lambda two and so
00:25:46.540 --> 00:25:47.130
on.
00:25:47.130 --> 00:25:50.355
Here go these
eigenvectors transposed.
00:25:54.870 --> 00:25:59.640
And what happens if I actually
do out that multiplication?
00:25:59.640 --> 00:26:03.630
Do you see what will happen?
00:26:03.630 --> 00:26:07.700
There's lambda one
times q1 transpose.
00:26:07.700 --> 00:26:11.850
So the first row here is
just lambda one q1 transpose.
00:26:11.850 --> 00:26:15.650
If I multiply
column times row --
00:26:15.650 --> 00:26:17.850
you remember I could do that?
00:26:17.850 --> 00:26:24.180
When I multiply matrixes, I can
multiply columns times rows?
00:26:24.180 --> 00:26:27.130
So when I do that, I
get lambda one and then
00:26:27.130 --> 00:26:31.900
the column and then
the row and then
00:26:31.900 --> 00:26:34.885
lambda two and then
the column and the row.
00:26:41.770 --> 00:26:46.980
Every symmetric matrix
breaks up into these pieces.
00:26:46.980 --> 00:26:54.950
So these pieces have real
lambdas and they have these
00:26:54.950 --> 00:26:56.833
Eigen -- these
orthonormal eigenvectors.
00:27:00.490 --> 00:27:04.560
And, maybe you even could
tell me what kind of a matrix
00:27:04.560 --> 00:27:07.370
have I got there?
00:27:07.370 --> 00:27:12.750
Suppose I take a unit
vector times its transpose?
00:27:12.750 --> 00:27:17.620
So column times row,
I'm getting a matrix.
00:27:17.620 --> 00:27:22.040
That's a matrix
with a special name.
00:27:22.040 --> 00:27:24.290
What's it's -- what
kind of a matrix is it?
00:27:24.290 --> 00:27:27.860
We've seen those matrixes,
now, in chapter four.
00:27:27.860 --> 00:27:32.440
It's -- is A A transpose
with a unit vector,
00:27:32.440 --> 00:27:36.210
so I don't have to
divide by A transpose A.
00:27:36.210 --> 00:27:40.580
That matrix is a
projection matrix.
00:27:40.580 --> 00:27:41.920
That's a projection matrix.
00:27:41.920 --> 00:27:46.550
It's symmetric and if I square
it there'll be another --
00:27:46.550 --> 00:27:50.040
there'll be a q1 transpose
q1, which is one.
00:27:50.040 --> 00:27:53.820
So I'll get that
matrix back again.
00:27:53.820 --> 00:27:57.070
Every -- so every
symmetric matrix --
00:27:57.070 --> 00:28:06.740
every symmetric matrix
is a combination of --
00:28:06.740 --> 00:28:14.230
of mutually perpendicular --
so perpendicular projection
00:28:14.230 --> 00:28:15.500
matrixes.
00:28:15.500 --> 00:28:17.442
Projection matrixes.
00:28:20.610 --> 00:28:21.530
Okay.
00:28:21.530 --> 00:28:23.940
That's another way
that people like
00:28:23.940 --> 00:28:27.770
to think of the
spectral theorem,
00:28:27.770 --> 00:28:31.790
that every symmetric matrix
can be broken up that way.
00:28:31.790 --> 00:28:35.220
That -- I guess
at this moment --
00:28:35.220 --> 00:28:36.800
first I haven't done an example.
00:28:36.800 --> 00:28:41.700
I could create a symmetric
matrix, check that it's --
00:28:41.700 --> 00:28:44.430
find its eigenvalues,
they would come out real,
00:28:44.430 --> 00:28:47.710
find its eigenvectors, they
would come out perpendicular
00:28:47.710 --> 00:28:51.230
and you would see it in numbers,
but maybe I'll leave it here
00:28:51.230 --> 00:28:54.650
for the moment in letters.
00:28:54.650 --> 00:28:58.520
Oh, I -- maybe I will do it
with numbers, for this reason.
00:28:58.520 --> 00:29:02.720
Because there's one
more remarkable fact.
00:29:02.720 --> 00:29:05.730
Can I just put this
further great fact
00:29:05.730 --> 00:29:07.890
about symmetric
matrixes on the board?
00:29:11.720 --> 00:29:13.560
When I have
symmetric matrixes, I
00:29:13.560 --> 00:29:18.140
know their eigenvalues are
So then I can get interested
00:29:18.140 --> 00:29:22.040
in the question are they
positive real. or negative?
00:29:22.040 --> 00:29:23.980
And you remember why
that's important.
00:29:23.980 --> 00:29:27.740
For differential equations,
that decides between instability
00:29:27.740 --> 00:29:29.920
and stability.
00:29:29.920 --> 00:29:32.620
So I'm -- after I
know they're real,
00:29:32.620 --> 00:29:34.970
then the next question
is are they positive,
00:29:34.970 --> 00:29:37.480
are they negative?
00:29:37.480 --> 00:29:44.140
And I hate to have to compute
those eigenvalues to answer
00:29:44.140 --> 00:29:46.120
that question, right?
00:29:46.120 --> 00:29:49.120
Because computing the
eigenvalues of a symmetric
00:29:49.120 --> 00:29:51.230
matrix of order let's say 50 --
00:29:51.230 --> 00:29:53.930
compute its 50 eigenvalues --
00:29:53.930 --> 00:29:58.630
is a job.
00:29:58.630 --> 00:30:04.050
I mean, by pencil and paper
it's a lifetime's job.
00:30:04.050 --> 00:30:11.480
I mean, which -- and in fact,
a few years ago -- well, say,
00:30:11.480 --> 00:30:17.660
20 years ago, or 30, nobody
really knew how to do it.
00:30:17.660 --> 00:30:22.140
I mean, so, like, science
was stuck on this problem.
00:30:22.140 --> 00:30:24.810
If you have a matrix
of order 50 or 100,
00:30:24.810 --> 00:30:27.320
how do you find its eigenvalues?
00:30:27.320 --> 00:30:29.530
Numerically, now,
I'm just saying,
00:30:29.530 --> 00:30:34.330
because pencil and paper is --
we're going to run out of time
00:30:34.330 --> 00:30:36.380
or paper or something
before we get it.
00:30:38.970 --> 00:30:41.840
Well -- and you
might think, okay,
00:30:41.840 --> 00:30:47.850
get Matlab to compute the
determinant of lambda minus A,
00:30:47.850 --> 00:30:52.330
A minus lambda I, this
polynomial of 50th degree,
00:30:52.330 --> 00:30:53.465
and then find the roots.
00:30:56.230 --> 00:30:59.340
Matlab will do it,
but it will complain,
00:30:59.340 --> 00:31:04.920
because it's a very bad way
to find the eigenvalues.
00:31:04.920 --> 00:31:08.480
I'm sorry to be saying
this, because it's the way I
00:31:08.480 --> 00:31:10.220
taught you to do it, right?
00:31:10.220 --> 00:31:12.000
I taught you to
find the eigenvalues
00:31:12.000 --> 00:31:14.830
by doing that
determinant and taking
00:31:14.830 --> 00:31:16.790
the roots of that polynomial.
00:31:16.790 --> 00:31:20.030
But now I'm saying, okay,
I really meant that for two
00:31:20.030 --> 00:31:21.820
by twos and three
by threes but I
00:31:21.820 --> 00:31:24.680
didn't mean you to
do it on a 50 by 50
00:31:24.680 --> 00:31:27.980
and you're not too
unhappy, probably,
00:31:27.980 --> 00:31:29.620
because you didn't
want to do it.
00:31:29.620 --> 00:31:36.650
But -- good, because it would
be a very unstable way --
00:31:36.650 --> 00:31:40.590
the 50 answers that would come
out would be highly unreliable.
00:31:40.590 --> 00:31:45.590
So, new ways are -- are
much better to find those 50
00:31:45.590 --> 00:31:46.310
eigenvalues.
00:31:46.310 --> 00:31:50.270
That's a -- that's a part
of numerical linear algebra.
00:31:50.270 --> 00:31:54.850
But here's the
remarkable fact --
00:31:54.850 --> 00:32:00.170
that Matlab would quite happily
find the 50 pivots, right?
00:32:00.170 --> 00:32:03.720
Now the pivots are not the
same as the eigenvalues.
00:32:03.720 --> 00:32:06.440
But here's the great thing.
00:32:06.440 --> 00:32:11.070
If I had a real matrix, I
could find those 50 pivots
00:32:11.070 --> 00:32:14.340
and I could see maybe
28 of them are positive
00:32:14.340 --> 00:32:15.680
and 22 are negative
00:32:15.680 --> 00:32:16.660
pivots.
00:32:16.660 --> 00:32:20.150
And I can compute those
safely and quickly.
00:32:20.150 --> 00:32:23.900
And the great fact is that 28
of the eigenvalues would be
00:32:23.900 --> 00:32:27.410
positive and 22
would be negative --
00:32:27.410 --> 00:32:31.740
that the sines of the pivots
-- so this is, like --
00:32:31.740 --> 00:32:34.780
I hope you think this --
this is kind of a nice thing,
00:32:34.780 --> 00:32:39.970
that the sines of the pivots --
00:32:39.970 --> 00:32:42.650
for symmetric, I'm always
talking about symmetric
00:32:42.650 --> 00:32:43.700
matrixes --
00:32:43.700 --> 00:32:45.890
so I'm really, like,
trying to convince you
00:32:45.890 --> 00:32:50.100
that symmetric matrixes
are better than the rest.
00:32:50.100 --> 00:32:58.700
So the sines of the pivots
are same as the sines
00:32:58.700 --> 00:33:02.070
of the eigenvalues.
00:33:02.070 --> 00:33:04.780
The same number.
00:33:04.780 --> 00:33:10.490
The number of pivots
greater than zero,
00:33:10.490 --> 00:33:16.120
the number of positive
pivots is equal to the number
00:33:16.120 --> 00:33:21.330
of positive eigenvalues.
00:33:21.330 --> 00:33:25.750
So that, actually, is a very
useful -- that gives you a g-
00:33:25.750 --> 00:33:31.340
a good start on a decent
way to compute eigenvalues,
00:33:31.340 --> 00:33:33.470
because you can
narrow them down,
00:33:33.470 --> 00:33:35.410
you can find out how
many are positive,
00:33:35.410 --> 00:33:37.490
how many are negative.
00:33:37.490 --> 00:33:42.680
Then you could shift the matrix
by seven times the identity.
00:33:42.680 --> 00:33:46.420
That would shift all the
eigenvalues by seven.
00:33:46.420 --> 00:33:48.500
Then you could take the
pivots of that matrix
00:33:48.500 --> 00:33:52.700
and you would know how many
eigenvalues of the original
00:33:52.700 --> 00:33:53.760
were above seven and
00:33:53.760 --> 00:33:54.870
below seven.
00:33:54.870 --> 00:33:59.070
So this -- this neat
little theorem, that,
00:33:59.070 --> 00:34:05.490
symmetric matrixes have this
connection between the --
00:34:05.490 --> 00:34:09.150
nobody's mixing up and thinking
the pivots are the eigenvalues
00:34:09.150 --> 00:34:10.820
--
00:34:10.820 --> 00:34:13.050
I mean, the only
thing I can think of
00:34:13.050 --> 00:34:15.880
is the product of
the pivots equals
00:34:15.880 --> 00:34:19.210
the product of the
eigenvalues, why is that?
00:34:19.210 --> 00:34:21.139
So if I asked you for
the reason on that,
00:34:21.139 --> 00:34:24.679
why is the product of the
pivots for a symmetric matrix
00:34:24.679 --> 00:34:27.639
the same as the product
of the eigenvalues?
00:34:27.639 --> 00:34:34.500
Because they both
equal the determinant.
00:34:34.500 --> 00:34:35.000
Right.
00:34:35.000 --> 00:34:37.060
The product of the pivots
gives the determinant
00:34:37.060 --> 00:34:40.710
if no row exchanges, the
product of the eigenvalues
00:34:40.710 --> 00:34:42.340
always gives the determinant.
00:34:42.340 --> 00:34:47.100
So -- so the products -- but
that doesn't tell you anything
00:34:47.100 --> 00:34:51.020
about the 50 individual
ones, which this does.
00:34:51.020 --> 00:34:51.810
Okay.
00:34:51.810 --> 00:34:57.566
So that's -- those are essential
facts about symmetric matrixes.
00:34:57.566 --> 00:34:58.066
Okay.
00:35:00.930 --> 00:35:06.410
Now I -- I said in the -- in
the lecture description that I
00:35:06.410 --> 00:35:13.680
would take the last minutes
to start on positive definite
00:35:13.680 --> 00:35:15.970
matrixes, because
we're right there,
00:35:15.970 --> 00:35:22.506
we're ready to say what's
a positive definite matrix?
00:35:31.740 --> 00:35:34.470
It's symmetric, first of all.
00:35:34.470 --> 00:35:37.090
On -- always I will
mean symmetric.
00:35:40.090 --> 00:35:43.690
So this is the -- this is
the next section of the book.
00:35:43.690 --> 00:35:45.420
It's about this --
00:35:45.420 --> 00:35:50.030
if symmetric matrixes are
good, which was, like,
00:35:50.030 --> 00:35:54.020
the point of my lecture
so far, then positive,
00:35:54.020 --> 00:35:57.650
definite matrixes are --
00:35:57.650 --> 00:36:03.180
a subclass that are
excellent, okay.
00:36:03.180 --> 00:36:05.430
Just the greatest.
00:36:05.430 --> 00:36:07.380
so what are they?
00:36:07.380 --> 00:36:10.930
They're matrixes --
they're symmetric matrixes,
00:36:10.930 --> 00:36:13.240
so all their
eigenvalues are real.
00:36:13.240 --> 00:36:15.420
You can guess what they are.
00:36:15.420 --> 00:36:20.070
These are symmetric
matrixes with all --
00:36:20.070 --> 00:36:21.190
the eigenvalues are --
00:36:25.790 --> 00:36:27.270
okay, tell me what to write.
00:36:31.240 --> 00:36:34.040
What -- well, it --
it's hinted, of course,
00:36:34.040 --> 00:36:36.200
by the name for these things.
00:36:36.200 --> 00:36:39.750
All the eigenvalues
are positive.
00:36:39.750 --> 00:36:40.250
Okay.
00:36:45.080 --> 00:36:46.900
Tell me about the pivots.
00:36:46.900 --> 00:36:50.200
We can check the eigenvalues
or we can check the pivots.
00:36:50.200 --> 00:36:53.270
All the pivots are what?
00:36:58.430 --> 00:36:59.730
And then I'll --
00:36:59.730 --> 00:37:01.230
then I'll finally
give an example.
00:37:01.230 --> 00:37:04.460
I feel awful that I have got
to this point in the lecture
00:37:04.460 --> 00:37:05.920
and I haven't given you a single
00:37:05.920 --> 00:37:06.790
example.
00:37:06.790 --> 00:37:08.690
So let me give you one.
00:37:08.690 --> 00:37:13.880
Five three two two.
00:37:13.880 --> 00:37:17.460
That's symmetric, fine.
00:37:17.460 --> 00:37:21.620
It's eigenvalues
are real, for sure.
00:37:21.620 --> 00:37:27.850
But more than that, I know the
sines of those eigenvalues.
00:37:27.850 --> 00:37:32.440
And also I know the
sines of those pivots,
00:37:32.440 --> 00:37:34.430
so what's the deal
with the pivots?
00:37:34.430 --> 00:37:39.800
The Ei- if the eigenvalues are
all positive and if this little
00:37:39.800 --> 00:37:44.240
fact is true that the pivots
and eigenvalues have the same
00:37:44.240 --> 00:37:48.680
sines, then this must be true
-- all the pivots are positive.
00:37:51.450 --> 00:37:54.330
And that's the good way to test.
00:37:54.330 --> 00:37:56.090
This is the good
test, because I can --
00:37:56.090 --> 00:37:59.090
what are the pivots
for that matrix?
00:37:59.090 --> 00:38:02.610
The pivots for that
matrix are five.
00:38:02.610 --> 00:38:08.840
So pivots are five and
what's the second pivot?
00:38:08.840 --> 00:38:13.570
Have we, like, noticed the
formula for the second pivot
00:38:13.570 --> 00:38:14.560
in a matrix?
00:38:18.332 --> 00:38:19.790
It doesn't necessarily
-- you know,
00:38:19.790 --> 00:38:22.200
it may come out a
fraction for sure,
00:38:22.200 --> 00:38:24.200
but what is that fraction?
00:38:24.200 --> 00:38:25.070
Can you tell me?
00:38:25.070 --> 00:38:30.070
Well, here, the product of
the pivots is the determinant.
00:38:30.070 --> 00:38:31.790
What's the determinant
of this matrix?
00:38:34.840 --> 00:38:36.460
Eleven?
00:38:36.460 --> 00:38:41.180
So the second pivot must
be eleven over five,
00:38:41.180 --> 00:38:44.600
so that the product is eleven.
00:38:44.600 --> 00:38:47.430
They're both positive.
00:38:47.430 --> 00:38:50.150
Then I know that the
eigenvalues of that matrix
00:38:50.150 --> 00:38:51.820
are both positive.
00:38:51.820 --> 00:38:53.140
What are the eigenvalues?
00:38:53.140 --> 00:38:55.550
Well, I've got to take
the roots of -- you know,
00:38:55.550 --> 00:38:57.770
do I put in a minus lambda?
00:38:57.770 --> 00:39:03.800
You mentally do this -- lambda
squared minus how many lambdas?
00:39:03.800 --> 00:39:04.310
Eight?
00:39:04.310 --> 00:39:04.810
Right.
00:39:04.810 --> 00:39:07.510
Five and three, the
trace comes in there,
00:39:07.510 --> 00:39:11.410
plus what number comes here?
00:39:11.410 --> 00:39:14.185
The determinant, the
eleven, so I set that to
00:39:14.185 --> 00:39:14.685
zero.
00:39:17.190 --> 00:39:20.190
So the eigenvalues are --
00:39:20.190 --> 00:39:24.040
let's see, half of that is four,
look at that positive number,
00:39:24.040 --> 00:39:28.600
plus or minus the square
root of sixteen minus eleven,
00:39:28.600 --> 00:39:29.360
I think five.
00:39:32.480 --> 00:39:35.450
The eigenvalues -- well, two
by two they're not so terrible,
00:39:35.450 --> 00:39:37.750
but they're not so perfect.
00:39:37.750 --> 00:39:40.235
Pivots are really simple.
00:39:44.580 --> 00:39:48.370
And this is a -- this is the
family of matrixes that you
00:39:48.370 --> 00:39:51.180
really want in
differential equations,
00:39:51.180 --> 00:39:55.720
because you know the
sines of the eigenvalues,
00:39:55.720 --> 00:39:58.520
so you know the
stability or not.
00:39:58.520 --> 00:39:59.340
Okay.
00:39:59.340 --> 00:40:03.700
There's one other related
fact I can pop in here in --
00:40:03.700 --> 00:40:07.745
in the time available for
positive definite matrixes.
00:40:10.380 --> 00:40:14.737
The related fact is to ask
you about determinants.
00:40:14.737 --> 00:40:15.820
So what's the determinant?
00:40:24.990 --> 00:40:27.470
What can you tell
me if I -- remember,
00:40:27.470 --> 00:40:32.090
positive definite means all
eigenvalues are positive,
00:40:32.090 --> 00:40:34.710
all pivots are positive, so
what can you tell me about
00:40:34.710 --> 00:40:36.890
the determinant?
00:40:36.890 --> 00:40:40.240
It's positive, too.
00:40:40.240 --> 00:40:45.070
But somehow that --
that's not quite enough.
00:40:45.070 --> 00:40:50.730
Here -- here's a matrix
minus one minus three,
00:40:50.730 --> 00:40:54.330
what's the determinant
of that guy?
00:40:54.330 --> 00:40:55.880
It's positive, right?
00:40:55.880 --> 00:40:58.010
Is this a positive,
definite matrix?
00:40:58.010 --> 00:41:00.000
Are the pivots --
what are the pivots?
00:41:00.000 --> 00:41:02.230
Well, negative.
00:41:02.230 --> 00:41:03.400
What are the eigenvalues?
00:41:03.400 --> 00:41:05.470
Well, they're also the same.
00:41:05.470 --> 00:41:12.240
So somehow I don't just want
the determinant of the whole
00:41:12.240 --> 00:41:12.850
matrix.
00:41:12.850 --> 00:41:14.770
Here is eleven, that's great.
00:41:14.770 --> 00:41:16.540
Here the determinant
of the whole matrix
00:41:16.540 --> 00:41:20.220
is three, that's positive.
00:41:20.220 --> 00:41:26.450
I also -- I've got to check,
like, little sub-determinants,
00:41:26.450 --> 00:41:29.330
say maybe coming
down from the left.
00:41:29.330 --> 00:41:32.950
So the one by one and the two
by two have to be positive.
00:41:32.950 --> 00:41:36.840
So there -- that's
where I get the all.
00:41:36.840 --> 00:41:41.140
All -- can I call them
sub-determinants --
00:41:41.140 --> 00:41:43.400
are -- see, I have to --
00:41:43.400 --> 00:41:45.670
I need to make the thing plural.
00:41:45.670 --> 00:41:51.230
I need to test n things, not
just the big determinant.
00:41:51.230 --> 00:41:55.130
All sub-determinants
are positive.
00:41:55.130 --> 00:41:58.110
Then I'm okay.
00:41:58.110 --> 00:42:00.610
Then I'm okay.
00:42:00.610 --> 00:42:03.310
This passes the test.
00:42:03.310 --> 00:42:06.830
Five is positive and
eleven is positive.
00:42:06.830 --> 00:42:12.220
This fails the test because that
minus one there is negative.
00:42:12.220 --> 00:42:16.050
And then the big determinant
is positive three.
00:42:16.050 --> 00:42:18.800
So t- this --
00:42:18.800 --> 00:42:23.320
these -- this fact -- you see
that actually the course, like,
00:42:23.320 --> 00:42:24.110
coming together.
00:42:27.210 --> 00:42:29.000
And that's really my point now.
00:42:29.000 --> 00:42:33.850
In the next -- in this lecture
and particularly next Wednesday
00:42:33.850 --> 00:42:38.150
and Friday, the
course comes together.
00:42:38.150 --> 00:42:41.810
These pivots that we
met in the first week,
00:42:41.810 --> 00:42:46.180
these determinants that we met
in the middle of the course,
00:42:46.180 --> 00:42:50.810
these eigenvalues that
we met most recently --
00:42:50.810 --> 00:42:56.010
all matrixes are square here,
so coming together for square
00:42:56.010 --> 00:43:00.150
matrixes means these three
pieces come together and they
00:43:00.150 --> 00:43:05.210
come together in that
beautiful fact, that if --
00:43:05.210 --> 00:43:07.430
that all the -- that
if I have one of these,
00:43:07.430 --> 00:43:09.150
I have the others.
00:43:09.150 --> 00:43:10.140
That if I --
00:43:10.140 --> 00:43:12.180
but for symmetric matrixes.
00:43:12.180 --> 00:43:17.570
So that -- this will be the
positive definite section
00:43:17.570 --> 00:43:22.540
and then the real climax of the
course is to make everything
00:43:22.540 --> 00:43:27.150
come together for
n by n matrixes,
00:43:27.150 --> 00:43:30.330
not necessarily symmetric --
00:43:30.330 --> 00:43:33.070
bring everything
together there and that
00:43:33.070 --> 00:43:34.850
will be the final thing.
00:43:34.850 --> 00:43:35.490
Okay.
00:43:35.490 --> 00:43:38.970
So have a great
weekend and don't
00:43:38.970 --> 00:43:40.710
forget symmetric matrixes.
00:43:40.710 --> 00:43:42.260
Thanks.