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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?
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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 --
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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 --
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the complex eigenvalues come
in lambda and lambda bar pairs.
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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 --
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00:12:05,750 --> 00:12:06,480
this is a -- then
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again okay.
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00:12:08,250 --> 00:12:10,000
Ax equals lambda x bar
transpose x, right?
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But now I'm ready
to use symmetry.
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I'm ready -- so this
was all just mechanics.
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Now -- now comes the
moment to say, okay,
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if the matrix is this
from the right with x bar,
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I get x bar transpose
Ax bar symmetric,
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then this A transpose
is the same as A.
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00:12:29,030 --> 00:12:31,910
You see, at that moment
I used the assumption.
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00:12:31,910 --> 00:12:34,380
Now let me finish
the discussion.
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00:12:34,380 --> 00:12:37,610
Here -- here's the way I finish.
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00:12:37,610 --> 00:12:42,460
I look at this original equation
and I take the inner product.
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00:12:42,460 --> 00:12:45,170
I multiply both sides by --
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00:12:45,170 --> 00:12:46,680
oh, maybe I'll do
it with this one.
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00:12:49,260 --> 00:12:51,540
I take --
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00:12:51,540 --> 00:12:55,200
I multiply both sides
by x bar transpose.
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00:12:55,200 --> 00:12:58,670
x bar transpose Ax
bar equals lambda
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00:12:58,670 --> 00:13:03,620
bar x bar transpose x bar.
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00:13:03,620 --> 00:13:04,470
Okay, fine.
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00:13:08,330 --> 00:13:12,170
All right, now
what's the other one?
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00:13:12,170 --> 00:13:16,880
Oh, for the other one I'll
probably use this guy.
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00:13:16,880 --> 00:13:20,580
A- I happy about this?
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00:13:20,580 --> 00:13:21,080
No.
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00:13:21,080 --> 00:13:22,910
For some reason I'm not.
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00:13:22,910 --> 00:13:26,830
I'm -- I want to --
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00:13:26,830 --> 00:14:04,850
if I take the inner
product of Okay.
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00:14:24,690 --> 00:14:26,660
So that -- that
was -- that's fine.
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00:14:26,660 --> 00:14:30,230
That comes directly from that,
multiplying both sides by x bar
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transpose, but now
I'd like to get --
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00:14:33,860 --> 00:14:38,320
why do I have x bars over there?
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Oh, yes.
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Forget this.
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Okay.
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00:14:43,660 --> 00:14:45,820
On this one -- right.
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00:14:45,820 --> 00:14:49,010
On this one, I took it like
that, I multiply on the right
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00:14:49,010 --> 00:14:50,400
by x.
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That's the idea.
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00:14:54,080 --> 00:14:55,540
Okay.
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00:14:55,540 --> 00:15:02,150
Now why I happier with
this situation now?
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A proof is coming here.
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00:15:04,180 --> 00:15:10,140
Because I compare this
guy with this one.
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00:15:10,140 --> 00:15:12,570
And they have the
same left hand side.
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So they have the
same right hand side.
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00:15:14,410 --> 00:15:16,240
So comparing those two, can --
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00:15:16,240 --> 00:15:19,910
I'll raise the board to
do this comparison --
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00:15:19,910 --> 00:15:25,150
this thing, lambda
x bar transpose x
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00:15:25,150 --> 00:15:32,880
is equal to lambda
bar x bar transpose x.
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Okay.
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And the conclusion
I'm going to reach --
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00:15:38,190 --> 00:15:43,280
I -- I on the right track here?
251
00:15:43,280 --> 00:15:44,830
The conclusion
I'm going to reach
252
00:15:44,830 --> 00:15:47,060
is lambda equal lambda bar.
253
00:15:52,170 --> 00:15:55,330
I would have to track down the
other possibility that this --
254
00:15:55,330 --> 00:15:58,400
this thing is
zero, but let me --
255
00:15:58,400 --> 00:16:01,330
oh -- oh, yes, that's important.
256
00:16:01,330 --> 00:16:03,370
It's not zero.
257
00:16:03,370 --> 00:16:08,760
So once I know that this
isn't zero, I just cancel it
258
00:16:08,760 --> 00:16:11,220
and I learn that lambda
equals lambda bar.
259
00:16:11,220 --> 00:16:14,360
And so what can you -- do you --
260
00:16:14,360 --> 00:16:17,823
have you got the
reasoning altogether?
261
00:16:20,640 --> 00:16:24,130
What does this tell us?
262
00:16:24,130 --> 00:16:27,750
Lambda's an eigenvalue
of this symmetric matrix.
263
00:16:27,750 --> 00:16:30,300
We've just proved that
it equaled lambda bar,
264
00:16:30,300 --> 00:16:34,630
so we have just proved
that lambda is real,
265
00:16:34,630 --> 00:16:35,710
right?
266
00:16:35,710 --> 00:16:39,980
If, if a number is equal to
its own complex conjugate,
267
00:16:39,980 --> 00:16:42,290
then there's no
imaginary part at all.
268
00:16:42,290 --> 00:16:43,280
The number is real.
269
00:16:43,280 --> 00:16:45,435
So lambda is real.
270
00:16:48,260 --> 00:16:49,380
Good.
271
00:16:49,380 --> 00:16:51,740
Good.
272
00:16:51,740 --> 00:16:56,310
Now, what -- but it depended
on this little expression,
273
00:16:56,310 --> 00:16:59,040
on knowing that
that wasn't zero,
274
00:16:59,040 --> 00:17:03,960
so that I could cancel it out
-- so can we just take a second
275
00:17:03,960 --> 00:17:05,400
on that one?
276
00:17:05,400 --> 00:17:08,650
Because it's an
important quantity.
277
00:17:08,650 --> 00:17:11,160
x bar transpose x.
278
00:17:11,160 --> 00:17:18,280
Okay, now remember, as far
as we know, x is complex.
279
00:17:18,280 --> 00:17:20,650
So this is --
280
00:17:20,650 --> 00:17:25,260
here -- x is complex, x
has these components, x1,
281
00:17:25,260 --> 00:17:28,310
x2 down to xn.
282
00:17:28,310 --> 00:17:35,890
And x bar transpose, well, it's
transposed and it's conjugated,
283
00:17:35,890 --> 00:17:42,570
so that's x1 conjugated x2
conjugated up to xn conjugated.
284
00:17:42,570 --> 00:17:43,520
I'm -- I'm --
285
00:17:43,520 --> 00:17:46,390
I'm really reminding
you of crucial facts
286
00:17:46,390 --> 00:17:48,440
about complex numbers
that are going
287
00:17:48,440 --> 00:17:51,230
to come into the next
lecture as well as this one.
288
00:17:51,230 --> 00:17:57,700
So w- what can you tell
me about that product --
289
00:17:57,700 --> 00:18:01,150
I -- I guess what
I'm trying to say is,
290
00:18:01,150 --> 00:18:05,830
if I had a complex vector, this
would be the quantity I would
291
00:18:05,830 --> 00:18:06,330
--
292
00:18:06,330 --> 00:18:07,730
I would like.
293
00:18:07,730 --> 00:18:09,310
This is the quantity I like.
294
00:18:09,310 --> 00:18:13,390
I would take the vector times
its transpose -- now what --
295
00:18:13,390 --> 00:18:17,070
what happens usually if I take a
vector -- a -- a -- x transpose
296
00:18:17,070 --> 00:18:18,070
x?
297
00:18:18,070 --> 00:18:22,330
I mean, that's a quantity we
see all the time, x transpose x.
298
00:18:22,330 --> 00:18:25,200
That's the length
of x squared, right?
299
00:18:25,200 --> 00:18:28,350
That's this positive length
squared, it's Pythagoras,
300
00:18:28,350 --> 00:18:31,640
it's x1 squared plus
x2 squared and so on.
301
00:18:31,640 --> 00:18:36,100
Now our vector's complex,
and you see the effect?
302
00:18:36,100 --> 00:18:39,090
I'm conjugating
one of these guys.
303
00:18:39,090 --> 00:18:41,180
So now when I do
this multiplication,
304
00:18:41,180 --> 00:18:49,370
I have x1 bar times x1 and
x2 bar times x2 and so on.
305
00:18:49,370 --> 00:18:53,640
So this is an --
this is sum a+ib.
306
00:18:53,640 --> 00:18:57,960
And this is sum a-ib.
307
00:18:57,960 --> 00:19:02,440
I mean, what's the point here?
308
00:19:02,440 --> 00:19:05,450
What's the point -- when
I multiply a number by its
309
00:19:05,450 --> 00:19:11,480
conjugate, a complex number by
its conjugate, what do I get?
310
00:19:11,480 --> 00:19:16,140
I get a n- the -- the
imaginary part is gone.
311
00:19:16,140 --> 00:19:20,680
When I multiply a+ib by
its conjugate, what's --
312
00:19:20,680 --> 00:19:23,400
what's the result of that -- of
each of those separate little
313
00:19:23,400 --> 00:19:25,210
multiplications?
314
00:19:25,210 --> 00:19:29,970
There's an a squared and -- and
what -- how many -- what's --
315
00:19:29,970 --> 00:19:34,010
b squared comes in
with a plus or a minus?
316
00:19:34,010 --> 00:19:35,290
A plus.
317
00:19:35,290 --> 00:19:39,260
i times minus i is
a plus b squared.
318
00:19:39,260 --> 00:19:41,240
And what about the
imaginary part?
319
00:19:43,830 --> 00:19:46,770
Gone, right?
320
00:19:46,770 --> 00:19:49,050
An iab and a minus iab.
321
00:19:49,050 --> 00:19:52,870
So this -- this is
the right thing to do.
322
00:19:52,870 --> 00:20:00,600
If you want a decent answer,
then multiply numbers
323
00:20:00,600 --> 00:20:02,910
by their conjugates.
324
00:20:02,910 --> 00:20:08,900
Multiply vectors by the
conjugates of x transpose.
325
00:20:08,900 --> 00:20:13,750
So this quantity is positive,
this quantity is positive --
326
00:20:13,750 --> 00:20:17,460
the whole thing is positive
except for the zero vector
327
00:20:17,460 --> 00:20:23,010
and that allows me to know
that this is a positive number,
328
00:20:23,010 --> 00:20:27,220
which I safely cancel out
and I reach the conclusion.
329
00:20:27,220 --> 00:20:33,290
So actually, in this discussion
here, I've done two things.
330
00:20:33,290 --> 00:20:35,690
If I reached the
conclusion that lambda's
331
00:20:35,690 --> 00:20:38,510
real, which I wanted to do.
332
00:20:38,510 --> 00:20:41,600
But at the same time,
we sort of saw what
333
00:20:41,600 --> 00:20:43,790
to do if things were complex.
334
00:20:43,790 --> 00:20:48,490
If a vector is complex,
then it's x bar transpose x,
335
00:20:48,490 --> 00:20:55,350
this is its length squared.
336
00:20:55,350 --> 00:20:59,510
And as I said, the next
lecture Monday, we'll --
337
00:20:59,510 --> 00:21:03,220
we'll repeat that this is
the right thing and then do
338
00:21:03,220 --> 00:21:06,280
the right thing for
matrixes and all other --
339
00:21:06,280 --> 00:21:10,950
all other, complex
possibilities.
340
00:21:10,950 --> 00:21:12,070
Okay.
341
00:21:12,070 --> 00:21:17,070
But the main point, then,
is that the eigenvalues
342
00:21:17,070 --> 00:21:20,720
of a symmetric matrix, it
just -- do you -- do --
343
00:21:20,720 --> 00:21:23,620
where did we use
symmetry, by the way?
344
00:21:23,620 --> 00:21:24,860
We used it here, right?
345
00:21:24,860 --> 00:21:27,870
Let -- can I just --
346
00:21:27,870 --> 00:21:31,980
let -- suppose A was a complex.
347
00:21:31,980 --> 00:21:34,800
Suppose A had been
a complex number.
348
00:21:34,800 --> 00:21:37,230
Could -- could I have
made all this work?
349
00:21:37,230 --> 00:21:40,780
If A was a complex
number -- complex matrix,
350
00:21:40,780 --> 00:21:45,070
then here I should
have written A bar.
351
00:21:45,070 --> 00:21:47,600
I erased the bar because
I assumed A was real.
352
00:21:47,600 --> 00:21:50,560
But now let's suppose
for a moment it's not.
353
00:21:50,560 --> 00:21:55,260
Then when I took this
step, what should I have?
354
00:21:55,260 --> 00:21:56,710
What did I do on that step?
355
00:21:56,710 --> 00:21:57,820
I transposed.
356
00:21:57,820 --> 00:22:00,215
So I should have
A bar transpose.
357
00:22:03,560 --> 00:22:05,990
In the symmetric
case, that was A,
358
00:22:05,990 --> 00:22:08,590
and that's what made
everything work, right?
359
00:22:08,590 --> 00:22:12,720
This -- this led
immediately to that.
360
00:22:12,720 --> 00:22:17,840
This one led immediately to
this when the matrix was real,
361
00:22:17,840 --> 00:22:20,300
so that didn't matter,
and it was symmetric,
362
00:22:20,300 --> 00:22:21,790
so that didn't matter.
363
00:22:21,790 --> 00:22:23,600
Then I got A.
364
00:22:23,600 --> 00:22:26,760
But -- so now I
just get to ask you.
365
00:22:26,760 --> 00:22:30,900
Suppose the matrix
had been complex.
366
00:22:30,900 --> 00:22:35,620
What's the right equivalent
of sym- symmetry?
367
00:22:38,320 --> 00:22:41,480
So the good matrix --
so here, let me say --
368
00:22:41,480 --> 00:22:52,776
good matrixes -- by good I mean
real lambdas and perpendicular
369
00:22:52,776 --> 00:22:53,276
x-s.
370
00:22:57,310 --> 00:23:02,860
And tell me now, which
matrixes are good?
371
00:23:02,860 --> 00:23:04,800
If they're --
372
00:23:04,800 --> 00:23:08,070
If they're real
matrixes, the good ones
373
00:23:08,070 --> 00:23:10,790
are symmetric, because then
everything went through.
374
00:23:10,790 --> 00:23:13,010
The -- so the good --
375
00:23:13,010 --> 00:23:15,050
I'm saying now what's good.
376
00:23:15,050 --> 00:23:17,430
This is -- this is -- these
are the good matrixes.
377
00:23:17,430 --> 00:23:21,030
They have real eigenvalues,
perpendicular eigenvectors --
378
00:23:21,030 --> 00:23:27,910
good means A equal
A transpose if real.
379
00:23:27,910 --> 00:23:30,560
Then -- then that was
what -- our proof worked.
380
00:23:30,560 --> 00:23:35,690
But if A is complex, all -- our
proof will still work provided
381
00:23:35,690 --> 00:23:38,060
A bar transpose is A.
382
00:23:38,060 --> 00:23:41,740
Do you see what I'm saying?
383
00:23:41,740 --> 00:23:47,700
I'm saying if we have complex
matrixes and we want to say are
384
00:23:47,700 --> 00:23:51,330
they -- are they as good
as symmetric matrixes,
385
00:23:51,330 --> 00:23:56,750
then we should not only
transpose the thing,
386
00:23:56,750 --> 00:23:58,760
but conjugate it.
387
00:23:58,760 --> 00:24:00,800
Those are good matrixes.
388
00:24:00,800 --> 00:24:03,090
And of course,
the most important
389
00:24:03,090 --> 00:24:06,980
s- the most important
case is when they're real,
390
00:24:06,980 --> 00:24:09,410
this part doesn't
matter and I just have
391
00:24:09,410 --> 00:24:11,330
A equal A transpose symmetric.
392
00:24:11,330 --> 00:24:12,070
Do you -- I --
393
00:24:12,070 --> 00:24:15,260
I'll just repeat that.
394
00:24:15,260 --> 00:24:20,290
The good matrixes, if
complex, are these.
395
00:24:20,290 --> 00:24:23,590
If real, that doesn't
make any difference
396
00:24:23,590 --> 00:24:25,900
so I'm just saying symmetric.
397
00:24:25,900 --> 00:24:30,530
And of course, 99% of
examples and applications
398
00:24:30,530 --> 00:24:34,740
to the matrixes are real
and we don't have that
399
00:24:34,740 --> 00:24:38,230
and then symmetric
is the key property.
400
00:24:38,230 --> 00:24:40,950
Okay.
401
00:24:40,950 --> 00:24:48,170
So that -- that's, these main
facts and now let me just --
402
00:24:48,170 --> 00:24:53,690
let me just -- so that's
this x bar transpose x, okay.
403
00:24:53,690 --> 00:24:59,210
So I'll just, write it
once more in this form.
404
00:24:59,210 --> 00:25:05,370
So perpendicular orthonormal
eigenvectors, real eigenvalues,
405
00:25:05,370 --> 00:25:08,410
transposes of
orthonormal eigenvectors.
406
00:25:08,410 --> 00:25:13,690
That's the symmetric
case, A equal A transpose.
407
00:25:13,690 --> 00:25:15,320
Okay.
408
00:25:15,320 --> 00:25:18,030
Good.
409
00:25:18,030 --> 00:25:23,340
Actually, I'll even
take one more step here.
410
00:25:23,340 --> 00:25:25,860
Suppose -- I --
411
00:25:25,860 --> 00:25:29,350
I can break this down
to show you really
412
00:25:29,350 --> 00:25:34,000
what that says about
a symmetric matrix.
413
00:25:34,000 --> 00:25:35,180
I can break that down.
414
00:25:35,180 --> 00:25:40,110
Let me here -- here
go these eigenvectors.
415
00:25:40,110 --> 00:25:46,540
I -- here go these eigenvalues,
lambda one, lambda two and so
416
00:25:46,540 --> 00:25:47,130
on.
417
00:25:47,130 --> 00:25:50,355
Here go these
eigenvectors transposed.
418
00:25:54,870 --> 00:25:59,640
And what happens if I actually
do out that multiplication?
419
00:25:59,640 --> 00:26:03,630
Do you see what will happen?
420
00:26:03,630 --> 00:26:07,700
There's lambda one
times q1 transpose.
421
00:26:07,700 --> 00:26:11,850
So the first row here is
just lambda one q1 transpose.
422
00:26:11,850 --> 00:26:15,650
If I multiply
column times row --
423
00:26:15,650 --> 00:26:17,850
you remember I could do that?
424
00:26:17,850 --> 00:26:24,180
When I multiply matrixes, I can
multiply columns times rows?
425
00:26:24,180 --> 00:26:27,130
So when I do that, I
get lambda one and then
426
00:26:27,130 --> 00:26:31,900
the column and then
the row and then
427
00:26:31,900 --> 00:26:34,885
lambda two and then
the column and the row.
428
00:26:41,770 --> 00:26:46,980
Every symmetric matrix
breaks up into these pieces.
429
00:26:46,980 --> 00:26:54,950
So these pieces have real
lambdas and they have these
430
00:26:54,950 --> 00:26:56,833
Eigen -- these
orthonormal eigenvectors.
431
00:27:00,490 --> 00:27:04,560
And, maybe you even could
tell me what kind of a matrix
432
00:27:04,560 --> 00:27:07,370
have I got there?
433
00:27:07,370 --> 00:27:12,750
Suppose I take a unit
vector times its transpose?
434
00:27:12,750 --> 00:27:17,620
So column times row,
I'm getting a matrix.
435
00:27:17,620 --> 00:27:22,040
That's a matrix
with a special name.
436
00:27:22,040 --> 00:27:24,290
What's it's -- what
kind of a matrix is it?
437
00:27:24,290 --> 00:27:27,860
We've seen those matrixes,
now, in chapter four.
438
00:27:27,860 --> 00:27:32,440
It's -- is A A transpose
with a unit vector,
439
00:27:32,440 --> 00:27:36,210
so I don't have to
divide by A transpose A.
440
00:27:36,210 --> 00:27:40,580
That matrix is a
projection matrix.
441
00:27:40,580 --> 00:27:41,920
That's a projection matrix.
442
00:27:41,920 --> 00:27:46,550
It's symmetric and if I square
it there'll be another --
443
00:27:46,550 --> 00:27:50,040
there'll be a q1 transpose
q1, which is one.
444
00:27:50,040 --> 00:27:53,820
So I'll get that
matrix back again.
445
00:27:53,820 --> 00:27:57,070
Every -- so every
symmetric matrix --
446
00:27:57,070 --> 00:28:06,740
every symmetric matrix
is a combination of --
447
00:28:06,740 --> 00:28:14,230
of mutually perpendicular --
so perpendicular projection
448
00:28:14,230 --> 00:28:15,500
matrixes.
449
00:28:15,500 --> 00:28:17,442
Projection matrixes.
450
00:28:20,610 --> 00:28:21,530
Okay.
451
00:28:21,530 --> 00:28:23,940
That's another way
that people like
452
00:28:23,940 --> 00:28:27,770
to think of the
spectral theorem,
453
00:28:27,770 --> 00:28:31,790
that every symmetric matrix
can be broken up that way.
454
00:28:31,790 --> 00:28:35,220
That -- I guess
at this moment --
455
00:28:35,220 --> 00:28:36,800
first I haven't done an example.
456
00:28:36,800 --> 00:28:41,700
I could create a symmetric
matrix, check that it's --
457
00:28:41,700 --> 00:28:44,430
find its eigenvalues,
they would come out real,
458
00:28:44,430 --> 00:28:47,710
find its eigenvectors, they
would come out perpendicular
459
00:28:47,710 --> 00:28:51,230
and you would see it in numbers,
but maybe I'll leave it here
460
00:28:51,230 --> 00:28:54,650
for the moment in letters.
461
00:28:54,650 --> 00:28:58,520
Oh, I -- maybe I will do it
with numbers, for this reason.
462
00:28:58,520 --> 00:29:02,720
Because there's one
more remarkable fact.
463
00:29:02,720 --> 00:29:05,730
Can I just put this
further great fact
464
00:29:05,730 --> 00:29:07,890
about symmetric
matrixes on the board?
465
00:29:11,720 --> 00:29:13,560
When I have
symmetric matrixes, I
466
00:29:13,560 --> 00:29:18,140
know their eigenvalues are
So then I can get interested
467
00:29:18,140 --> 00:29:22,040
in the question are they
positive real. or negative?
468
00:29:22,040 --> 00:29:23,980
And you remember why
that's important.
469
00:29:23,980 --> 00:29:27,740
For differential equations,
that decides between instability
470
00:29:27,740 --> 00:29:29,920
and stability.
471
00:29:29,920 --> 00:29:32,620
So I'm -- after I
know they're real,
472
00:29:32,620 --> 00:29:34,970
then the next question
is are they positive,
473
00:29:34,970 --> 00:29:37,480
are they negative?
474
00:29:37,480 --> 00:29:44,140
And I hate to have to compute
those eigenvalues to answer
475
00:29:44,140 --> 00:29:46,120
that question, right?
476
00:29:46,120 --> 00:29:49,120
Because computing the
eigenvalues of a symmetric
477
00:29:49,120 --> 00:29:51,230
matrix of order let's say 50 --
478
00:29:51,230 --> 00:29:53,930
compute its 50 eigenvalues --
479
00:29:53,930 --> 00:29:58,630
is a job.
480
00:29:58,630 --> 00:30:04,050
I mean, by pencil and paper
it's a lifetime's job.
481
00:30:04,050 --> 00:30:11,480
I mean, which -- and in fact,
a few years ago -- well, say,
482
00:30:11,480 --> 00:30:17,660
20 years ago, or 30, nobody
really knew how to do it.
483
00:30:17,660 --> 00:30:22,140
I mean, so, like, science
was stuck on this problem.
484
00:30:22,140 --> 00:30:24,810
If you have a matrix
of order 50 or 100,
485
00:30:24,810 --> 00:30:27,320
how do you find its eigenvalues?
486
00:30:27,320 --> 00:30:29,530
Numerically, now,
I'm just saying,
487
00:30:29,530 --> 00:30:34,330
because pencil and paper is --
we're going to run out of time
488
00:30:34,330 --> 00:30:36,380
or paper or something
before we get it.
489
00:30:38,970 --> 00:30:41,840
Well -- and you
might think, okay,
490
00:30:41,840 --> 00:30:47,850
get Matlab to compute the
determinant of lambda minus A,
491
00:30:47,850 --> 00:30:52,330
A minus lambda I, this
polynomial of 50th degree,
492
00:30:52,330 --> 00:30:53,465
and then find the roots.
493
00:30:56,230 --> 00:30:59,340
Matlab will do it,
but it will complain,
494
00:30:59,340 --> 00:31:04,920
because it's a very bad way
to find the eigenvalues.
495
00:31:04,920 --> 00:31:08,480
I'm sorry to be saying
this, because it's the way I
496
00:31:08,480 --> 00:31:10,220
taught you to do it, right?
497
00:31:10,220 --> 00:31:12,000
I taught you to
find the eigenvalues
498
00:31:12,000 --> 00:31:14,830
by doing that
determinant and taking
499
00:31:14,830 --> 00:31:16,790
the roots of that polynomial.
500
00:31:16,790 --> 00:31:20,030
But now I'm saying, okay,
I really meant that for two
501
00:31:20,030 --> 00:31:21,820
by twos and three
by threes but I
502
00:31:21,820 --> 00:31:24,680
didn't mean you to
do it on a 50 by 50
503
00:31:24,680 --> 00:31:27,980
and you're not too
unhappy, probably,
504
00:31:27,980 --> 00:31:29,620
because you didn't
want to do it.
505
00:31:29,620 --> 00:31:36,650
But -- good, because it would
be a very unstable way --
506
00:31:36,650 --> 00:31:40,590
the 50 answers that would come
out would be highly unreliable.
507
00:31:40,590 --> 00:31:45,590
So, new ways are -- are
much better to find those 50
508
00:31:45,590 --> 00:31:46,310
eigenvalues.
509
00:31:46,310 --> 00:31:50,270
That's a -- that's a part
of numerical linear algebra.
510
00:31:50,270 --> 00:31:54,850
But here's the
remarkable fact --
511
00:31:54,850 --> 00:32:00,170
that Matlab would quite happily
find the 50 pivots, right?
512
00:32:00,170 --> 00:32:03,720
Now the pivots are not the
same as the eigenvalues.
513
00:32:03,720 --> 00:32:06,440
But here's the great thing.
514
00:32:06,440 --> 00:32:11,070
If I had a real matrix, I
could find those 50 pivots
515
00:32:11,070 --> 00:32:14,340
and I could see maybe
28 of them are positive
516
00:32:14,340 --> 00:32:15,680
and 22 are negative
517
00:32:15,680 --> 00:32:16,660
pivots.
518
00:32:16,660 --> 00:32:20,150
And I can compute those
safely and quickly.
519
00:32:20,150 --> 00:32:23,900
And the great fact is that 28
of the eigenvalues would be
520
00:32:23,900 --> 00:32:27,410
positive and 22
would be negative --
521
00:32:27,410 --> 00:32:31,740
that the sines of the pivots
-- so this is, like --
522
00:32:31,740 --> 00:32:34,780
I hope you think this --
this is kind of a nice thing,
523
00:32:34,780 --> 00:32:39,970
that the sines of the pivots --
524
00:32:39,970 --> 00:32:42,650
for symmetric, I'm always
talking about symmetric
525
00:32:42,650 --> 00:32:43,700
matrixes --
526
00:32:43,700 --> 00:32:45,890
so I'm really, like,
trying to convince you
527
00:32:45,890 --> 00:32:50,100
that symmetric matrixes
are better than the rest.
528
00:32:50,100 --> 00:32:58,700
So the sines of the pivots
are same as the sines
529
00:32:58,700 --> 00:33:02,070
of the eigenvalues.
530
00:33:02,070 --> 00:33:04,780
The same number.
531
00:33:04,780 --> 00:33:10,490
The number of pivots
greater than zero,
532
00:33:10,490 --> 00:33:16,120
the number of positive
pivots is equal to the number
533
00:33:16,120 --> 00:33:21,330
of positive eigenvalues.
534
00:33:21,330 --> 00:33:25,750
So that, actually, is a very
useful -- that gives you a g-
535
00:33:25,750 --> 00:33:31,340
a good start on a decent
way to compute eigenvalues,
536
00:33:31,340 --> 00:33:33,470
because you can
narrow them down,
537
00:33:33,470 --> 00:33:35,410
you can find out how
many are positive,
538
00:33:35,410 --> 00:33:37,490
how many are negative.
539
00:33:37,490 --> 00:33:42,680
Then you could shift the matrix
by seven times the identity.
540
00:33:42,680 --> 00:33:46,420
That would shift all the
eigenvalues by seven.
541
00:33:46,420 --> 00:33:48,500
Then you could take the
pivots of that matrix
542
00:33:48,500 --> 00:33:52,700
and you would know how many
eigenvalues of the original
543
00:33:52,700 --> 00:33:53,760
were above seven and
544
00:33:53,760 --> 00:33:54,870
below seven.
545
00:33:54,870 --> 00:33:59,070
So this -- this neat
little theorem, that,
546
00:33:59,070 --> 00:34:05,490
symmetric matrixes have this
connection between the --
547
00:34:05,490 --> 00:34:09,150
nobody's mixing up and thinking
the pivots are the eigenvalues
548
00:34:09,150 --> 00:34:10,820
--
549
00:34:10,820 --> 00:34:13,050
I mean, the only
thing I can think of
550
00:34:13,050 --> 00:34:15,880
is the product of
the pivots equals
551
00:34:15,880 --> 00:34:19,210
the product of the
eigenvalues, why is that?
552
00:34:19,210 --> 00:34:21,139
So if I asked you for
the reason on that,
553
00:34:21,139 --> 00:34:24,679
why is the product of the
pivots for a symmetric matrix
554
00:34:24,679 --> 00:34:27,639
the same as the product
of the eigenvalues?
555
00:34:27,639 --> 00:34:34,500
Because they both
equal the determinant.
556
00:34:34,500 --> 00:34:35,000
Right.
557
00:34:35,000 --> 00:34:37,060
The product of the pivots
gives the determinant
558
00:34:37,060 --> 00:34:40,710
if no row exchanges, the
product of the eigenvalues
559
00:34:40,710 --> 00:34:42,340
always gives the determinant.
560
00:34:42,340 --> 00:34:47,100
So -- so the products -- but
that doesn't tell you anything
561
00:34:47,100 --> 00:34:51,020
about the 50 individual
ones, which this does.
562
00:34:51,020 --> 00:34:51,810
Okay.
563
00:34:51,810 --> 00:34:57,566
So that's -- those are essential
facts about symmetric matrixes.
564
00:34:57,566 --> 00:34:58,066
Okay.
565
00:35:00,930 --> 00:35:06,410
Now I -- I said in the -- in
the lecture description that I
566
00:35:06,410 --> 00:35:13,680
would take the last minutes
to start on positive definite
567
00:35:13,680 --> 00:35:15,970
matrixes, because
we're right there,
568
00:35:15,970 --> 00:35:22,506
we're ready to say what's
a positive definite matrix?
569
00:35:31,740 --> 00:35:34,470
It's symmetric, first of all.
570
00:35:34,470 --> 00:35:37,090
On -- always I will
mean symmetric.
571
00:35:40,090 --> 00:35:43,690
So this is the -- this is
the next section of the book.
572
00:35:43,690 --> 00:35:45,420
It's about this --
573
00:35:45,420 --> 00:35:50,030
if symmetric matrixes are
good, which was, like,
574
00:35:50,030 --> 00:35:54,020
the point of my lecture
so far, then positive,
575
00:35:54,020 --> 00:35:57,650
definite matrixes are --
576
00:35:57,650 --> 00:36:03,180
a subclass that are
excellent, okay.
577
00:36:03,180 --> 00:36:05,430
Just the greatest.
578
00:36:05,430 --> 00:36:07,380
so what are they?
579
00:36:07,380 --> 00:36:10,930
They're matrixes --
they're symmetric matrixes,
580
00:36:10,930 --> 00:36:13,240
so all their
eigenvalues are real.
581
00:36:13,240 --> 00:36:15,420
You can guess what they are.
582
00:36:15,420 --> 00:36:20,070
These are symmetric
matrixes with all --
583
00:36:20,070 --> 00:36:21,190
the eigenvalues are --
584
00:36:25,790 --> 00:36:27,270
okay, tell me what to write.
585
00:36:31,240 --> 00:36:34,040
What -- well, it --
it's hinted, of course,
586
00:36:34,040 --> 00:36:36,200
by the name for these things.
587
00:36:36,200 --> 00:36:39,750
All the eigenvalues
are positive.
588
00:36:39,750 --> 00:36:40,250
Okay.
589
00:36:45,080 --> 00:36:46,900
Tell me about the pivots.
590
00:36:46,900 --> 00:36:50,200
We can check the eigenvalues
or we can check the pivots.
591
00:36:50,200 --> 00:36:53,270
All the pivots are what?
592
00:36:58,430 --> 00:36:59,730
And then I'll --
593
00:36:59,730 --> 00:37:01,230
then I'll finally
give an example.
594
00:37:01,230 --> 00:37:04,460
I feel awful that I have got
to this point in the lecture
595
00:37:04,460 --> 00:37:05,920
and I haven't given you a single
596
00:37:05,920 --> 00:37:06,790
example.
597
00:37:06,790 --> 00:37:08,690
So let me give you one.
598
00:37:08,690 --> 00:37:13,880
Five three two two.
599
00:37:13,880 --> 00:37:17,460
That's symmetric, fine.
600
00:37:17,460 --> 00:37:21,620
It's eigenvalues
are real, for sure.
601
00:37:21,620 --> 00:37:27,850
But more than that, I know the
sines of those eigenvalues.
602
00:37:27,850 --> 00:37:32,440
And also I know the
sines of those pivots,
603
00:37:32,440 --> 00:37:34,430
so what's the deal
with the pivots?
604
00:37:34,430 --> 00:37:39,800
The Ei- if the eigenvalues are
all positive and if this little
605
00:37:39,800 --> 00:37:44,240
fact is true that the pivots
and eigenvalues have the same
606
00:37:44,240 --> 00:37:48,680
sines, then this must be true
-- all the pivots are positive.
607
00:37:51,450 --> 00:37:54,330
And that's the good way to test.
608
00:37:54,330 --> 00:37:56,090
This is the good
test, because I can --
609
00:37:56,090 --> 00:37:59,090
what are the pivots
for that matrix?
610
00:37:59,090 --> 00:38:02,610
The pivots for that
matrix are five.
611
00:38:02,610 --> 00:38:08,840
So pivots are five and
what's the second pivot?
612
00:38:08,840 --> 00:38:13,570
Have we, like, noticed the
formula for the second pivot
613
00:38:13,570 --> 00:38:14,560
in a matrix?
614
00:38:18,332 --> 00:38:19,790
It doesn't necessarily
-- you know,
615
00:38:19,790 --> 00:38:22,200
it may come out a
fraction for sure,
616
00:38:22,200 --> 00:38:24,200
but what is that fraction?
617
00:38:24,200 --> 00:38:25,070
Can you tell me?
618
00:38:25,070 --> 00:38:30,070
Well, here, the product of
the pivots is the determinant.
619
00:38:30,070 --> 00:38:31,790
What's the determinant
of this matrix?
620
00:38:34,840 --> 00:38:36,460
Eleven?
621
00:38:36,460 --> 00:38:41,180
So the second pivot must
be eleven over five,
622
00:38:41,180 --> 00:38:44,600
so that the product is eleven.
623
00:38:44,600 --> 00:38:47,430
They're both positive.
624
00:38:47,430 --> 00:38:50,150
Then I know that the
eigenvalues of that matrix
625
00:38:50,150 --> 00:38:51,820
are both positive.
626
00:38:51,820 --> 00:38:53,140
What are the eigenvalues?
627
00:38:53,140 --> 00:38:55,550
Well, I've got to take
the roots of -- you know,
628
00:38:55,550 --> 00:38:57,770
do I put in a minus lambda?
629
00:38:57,770 --> 00:39:03,800
You mentally do this -- lambda
squared minus how many lambdas?
630
00:39:03,800 --> 00:39:04,310
Eight?
631
00:39:04,310 --> 00:39:04,810
Right.
632
00:39:04,810 --> 00:39:07,510
Five and three, the
trace comes in there,
633
00:39:07,510 --> 00:39:11,410
plus what number comes here?
634
00:39:11,410 --> 00:39:14,185
The determinant, the
eleven, so I set that to
635
00:39:14,185 --> 00:39:14,685
zero.
636
00:39:17,190 --> 00:39:20,190
So the eigenvalues are --
637
00:39:20,190 --> 00:39:24,040
let's see, half of that is four,
look at that positive number,
638
00:39:24,040 --> 00:39:28,600
plus or minus the square
root of sixteen minus eleven,
639
00:39:28,600 --> 00:39:29,360
I think five.
640
00:39:32,480 --> 00:39:35,450
The eigenvalues -- well, two
by two they're not so terrible,
641
00:39:35,450 --> 00:39:37,750
but they're not so perfect.
642
00:39:37,750 --> 00:39:40,235
Pivots are really simple.
643
00:39:44,580 --> 00:39:48,370
And this is a -- this is the
family of matrixes that you
644
00:39:48,370 --> 00:39:51,180
really want in
differential equations,
645
00:39:51,180 --> 00:39:55,720
because you know the
sines of the eigenvalues,
646
00:39:55,720 --> 00:39:58,520
so you know the
stability or not.
647
00:39:58,520 --> 00:39:59,340
Okay.
648
00:39:59,340 --> 00:40:03,700
There's one other related
fact I can pop in here in --
649
00:40:03,700 --> 00:40:07,745
in the time available for
positive definite matrixes.
650
00:40:10,380 --> 00:40:14,737
The related fact is to ask
you about determinants.
651
00:40:14,737 --> 00:40:15,820
So what's the determinant?
652
00:40:24,990 --> 00:40:27,470
What can you tell
me if I -- remember,
653
00:40:27,470 --> 00:40:32,090
positive definite means all
eigenvalues are positive,
654
00:40:32,090 --> 00:40:34,710
all pivots are positive, so
what can you tell me about
655
00:40:34,710 --> 00:40:36,890
the determinant?
656
00:40:36,890 --> 00:40:40,240
It's positive, too.
657
00:40:40,240 --> 00:40:45,070
But somehow that --
that's not quite enough.
658
00:40:45,070 --> 00:40:50,730
Here -- here's a matrix
minus one minus three,
659
00:40:50,730 --> 00:40:54,330
what's the determinant
of that guy?
660
00:40:54,330 --> 00:40:55,880
It's positive, right?
661
00:40:55,880 --> 00:40:58,010
Is this a positive,
definite matrix?
662
00:40:58,010 --> 00:41:00,000
Are the pivots --
what are the pivots?
663
00:41:00,000 --> 00:41:02,230
Well, negative.
664
00:41:02,230 --> 00:41:03,400
What are the eigenvalues?
665
00:41:03,400 --> 00:41:05,470
Well, they're also the same.
666
00:41:05,470 --> 00:41:12,240
So somehow I don't just want
the determinant of the whole
667
00:41:12,240 --> 00:41:12,850
matrix.
668
00:41:12,850 --> 00:41:14,770
Here is eleven, that's great.
669
00:41:14,770 --> 00:41:16,540
Here the determinant
of the whole matrix
670
00:41:16,540 --> 00:41:20,220
is three, that's positive.
671
00:41:20,220 --> 00:41:26,450
I also -- I've got to check,
like, little sub-determinants,
672
00:41:26,450 --> 00:41:29,330
say maybe coming
down from the left.
673
00:41:29,330 --> 00:41:32,950
So the one by one and the two
by two have to be positive.
674
00:41:32,950 --> 00:41:36,840
So there -- that's
where I get the all.
675
00:41:36,840 --> 00:41:41,140
All -- can I call them
sub-determinants --
676
00:41:41,140 --> 00:41:43,400
are -- see, I have to --
677
00:41:43,400 --> 00:41:45,670
I need to make the thing plural.
678
00:41:45,670 --> 00:41:51,230
I need to test n things, not
just the big determinant.
679
00:41:51,230 --> 00:41:55,130
All sub-determinants
are positive.
680
00:41:55,130 --> 00:41:58,110
Then I'm okay.
681
00:41:58,110 --> 00:42:00,610
Then I'm okay.
682
00:42:00,610 --> 00:42:03,310
This passes the test.
683
00:42:03,310 --> 00:42:06,830
Five is positive and
eleven is positive.
684
00:42:06,830 --> 00:42:12,220
This fails the test because that
minus one there is negative.
685
00:42:12,220 --> 00:42:16,050
And then the big determinant
is positive three.
686
00:42:16,050 --> 00:42:18,800
So t- this --
687
00:42:18,800 --> 00:42:23,320
these -- this fact -- you see
that actually the course, like,
688
00:42:23,320 --> 00:42:24,110
coming together.
689
00:42:27,210 --> 00:42:29,000
And that's really my point now.
690
00:42:29,000 --> 00:42:33,850
In the next -- in this lecture
and particularly next Wednesday
691
00:42:33,850 --> 00:42:38,150
and Friday, the
course comes together.
692
00:42:38,150 --> 00:42:41,810
These pivots that we
met in the first week,
693
00:42:41,810 --> 00:42:46,180
these determinants that we met
in the middle of the course,
694
00:42:46,180 --> 00:42:50,810
these eigenvalues that
we met most recently --
695
00:42:50,810 --> 00:42:56,010
all matrixes are square here,
so coming together for square
696
00:42:56,010 --> 00:43:00,150
matrixes means these three
pieces come together and they
697
00:43:00,150 --> 00:43:05,210
come together in that
beautiful fact, that if --
698
00:43:05,210 --> 00:43:07,430
that all the -- that
if I have one of these,
699
00:43:07,430 --> 00:43:09,150
I have the others.
700
00:43:09,150 --> 00:43:10,140
That if I --
701
00:43:10,140 --> 00:43:12,180
but for symmetric matrixes.
702
00:43:12,180 --> 00:43:17,570
So that -- this will be the
positive definite section
703
00:43:17,570 --> 00:43:22,540
and then the real climax of the
course is to make everything
704
00:43:22,540 --> 00:43:27,150
come together for
n by n matrixes,
705
00:43:27,150 --> 00:43:30,330
not necessarily symmetric --
706
00:43:30,330 --> 00:43:33,070
bring everything
together there and that
707
00:43:33,070 --> 00:43:34,850
will be the final thing.
708
00:43:34,850 --> 00:43:35,490
Okay.
709
00:43:35,490 --> 00:43:38,970
So have a great
weekend and don't
710
00:43:38,970 --> 00:43:40,710
forget symmetric matrixes.
711
00:43:40,710 --> 00:43:42,260
Thanks.