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OK, here we go with, quiz
review for the third quiz that's
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coming on Friday.
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So, one key point
is that the quiz
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covers through chapter six.
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Chapter seven on
linear transformations
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will appear on the final
exam, but not on the quiz.
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So I won't review linear
transformations today,
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but they'll come into the
full course review on the very
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last lecture.
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So today, I'm
reviewing chapter six,
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and I'm going to
take some old exams,
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and I'm always ready
to answer questions.
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And I thought, kind of help
our memories if I write down
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the main topics in chapter six.
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So, already, on
the previous quiz,
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we knew how to find
eigenvalues and eigenvectors.
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Well, we knew how to find them
by that determinant of A minus
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lambda I equals zero.
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But, of course, there
could be shortcuts.
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There could be, like,
useful information
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about the eigenvalues that
we can speed things up with.
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OK.
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Then, the new stuff starts out
with a differential equation,
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so I'll do a problem.
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I'll do a differential
equation problem first.
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What's special about
symmetric matrices?
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Can we just say that in words?
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I'd better write
it down, though.
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What's special about
symmetric matrices?
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Their eigenvalues are real.
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The eigenvalues of a symmetric
matrix always come out real,
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and there always are
enough eigenvectors.
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Even if there are
repeated eigenvalues,
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there are enough
eigenvectors, and we
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can choose those eigenvectors
to be orthogonal.
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So if A equals A
transposed, the big fact
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will be that we
can diagonalize it,
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and those eigenvector
matrix, with the eigenvectors
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in the column, can be
an orthogonal matrix.
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So we get a Q
lambda Q transpose.
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That, in three symbols,
expresses a wonderful fact,
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a fundamental fact for
symmetric matrices.
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OK.
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Then, we went beyond
that fact to ask
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about positive
definite matrices, when
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the eigenvalues were positive.
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I'll do an example of that.
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Now we've left symmetry.
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Similar matrices are
any square matrices,
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but two matrices are similar
if they're related that way.
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And what's the key point
about similar matrices?
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Somehow, those matrices
are representing
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the same thing in
different basis,
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in chapter seven language.
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In chapter six language, what's
up with these similar matrices?
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What's the key fact,
the key positive fact
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about similar matrices?
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They have the same eigenvalues.
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Same eigenvalues.
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So if one of them grows,
the other one grows.
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If one of them decays to zero,
the other one decays to zero.
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Powers of A will look
like powers of B,
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because powers of
A and powers of B
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only differ by an M
inverse and an M way on the
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outside.
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So if these are similar,
then B to the k-th power
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is M inverse A to
the k-th power M.
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And that's why I
say, eh, this M, it
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does change the
eigenvectors, but it
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doesn't change the eigenvalues.
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So same lambdas.
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And then, finally, I've
got to review the point
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about the SVD, the Singular
Value Decomposition.
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OK.
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So that's what this
quiz has got to cover,
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and now I'll just take
problems from earlier exams,
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starting with a
differential equation.
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OK.
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And always ready for questions.
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So here is an exam
from about the year
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zero, and it has
a three by three.
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So that was --
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but it's a pretty
special-looking matrix,
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it's got zeroes on the diagonal,
it's got minus ones above,
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and it's got plus
ones like that.
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So that's the matrix A.
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OK.
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Step one is, well,
I want to solve that
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equation.
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I want to find the
general solution.
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I haven't given you
a u(0) here, so I'm
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looking for the
general solution,
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so now what's the form
of the general solution?
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With three arbitrary
constants going
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to be inside it, because
those will be used
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to match the initial condition.
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So the general
form is u at time t
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is some multiple of the
first special solution.
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The first special solution will
be growing like the eigenvalue,
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and it's the eigenvector.
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So that's a pure exponential
solution, just staying
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with that eigenvector.
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Of course, I haven't
found, yet, the eigenvalues
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and eigenvectors.
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That's, normally, the first job.
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Now, there will be second
one, growing like e
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to the lambda two, and a
third one growing like e
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to the lambda three.
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So we're all done --
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well, we haven't
done anything yet,
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actually.
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I've got to find the
eigenvalues and eigenvectors,
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and then I would match u(0)
by choosing the right three
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constants.
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OK.
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So now I ask -- ask you
about the eigenvalues
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and eigenvectors, and you look
at this matrix and what do you
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see in that matrix?
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Um, well, I guess we might
ask ourselves right away,
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is it singular?
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Is it singular?
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Because, if so, then we
really have a head start,
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we know one of the
eigenvalues is zero.
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Is that matrix singular?
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Eh, I don't know, do you
take the determinant to
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find out?
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Or maybe you look at the
first row and third row
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and say, hey, the
first row and third row
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are just opposite signs,
they're linear-dependent?
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The first column and third
column are dependent -- it's
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singular.
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So one eigenvalue is zero.
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Let's make that lambda one.
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Lambda one, then, will be zero.
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OK.
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Now we've got a couple of
other eigenvalues to find,
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and, I suppose the
simplest way is
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to look at A minus
lambda I So let
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me just put minus lambda
in here, minus ones above,
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ones below.
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But, actually, before
I do it, that matrix
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is not symmetric,
for sure, right?
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In fact, it's the very
opposite of symmetric.
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That matrix A transpose, how
is A transpose connected to A?
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It's negative A.
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It's an anti-symmetric
matrix, skew-symmetric matrix.
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And we've met, maybe,
a two-by-two example
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of skew-symmetric
matrices, and let
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me just say, what's the
deal with their eigenvalues?
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They're pure imaginary.
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They'll be on the
imaginary axis,
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there be some
multiple of I if it's
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an anti-symmetric,
skew-symmetric matrix.
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So I'm looking for
multiples of I,
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and of course,
that's zero times I,
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that's on the imaginary axis,
but maybe I just do it out,
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here.
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Lambda cubed.
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well, maybe that's
minus lambda cubed,
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and then a zero and a zero.
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Zero, and then maybe I
have a plus a lambda,
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and another plus lambda, but
those go with a minus sign.
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Am I getting minus two
lambda equals zero?
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So.
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So I'm solving lambda cube
plus two lambda equals zero.
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So one root factors out
lambda, and the the rest
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is lambda squared plus two.
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OK.
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This is going the
way we expect, right?
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Because this gives the root
lambda equals zero, and gives
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the other two roots, which
are lambda equal what?
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The solutions of when
is lambda squared
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plus two equals zero then the
eigenvalues those guys, what
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are they?
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They're a multiple of i,
they're just square root of two
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i.
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When I set this
equals to zero, I
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have lambda squared equal
to minus two, right?
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To make that zero?
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And the roots are
square root of two i
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and minus the square
root of two i.
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So now I know what those are.
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I'll put those in, now.
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Either the zero t is just a one.
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That's just a one.
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This is square root of
two I and this is minus
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square root of two I.
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So, is the solution
decaying to zero?
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Is this a completely
stable problem
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where the solution
is going to zero?
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No.
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In fact, all these things
are staying the same size.
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This thing is getting
multiplied by this number.
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e to the I something t, that's
a number that has magnitude one,
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and sort of wanders
around the unit circle.
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Same for this.
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So that the solution doesn't
blow up, and it doesn't go to
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zero.
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OK.
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And to find out
what it actually is,
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we would have to plug
in initial conditions.
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But actually, the
next question I ask
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is, when does the solution
return to its initial value?
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I won't even say what's
the initial value.
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This is a case in which
I think this solution is
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periodic after.
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00:11:25,350 --> 00:11:31,820
At t equals zero, it
starts with c1, c2, and c3,
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00:11:31,820 --> 00:11:36,500
and then at some value of
t, it comes back to that.
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So that's a very
special question,
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00:11:38,440 --> 00:11:40,280
Well, let's just
take three seconds,
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because that special question
isn't likely to be on the quiz.
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But it comes back
to the start, when?
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00:11:49,910 --> 00:11:55,230
Well, whenever we have e to
the two pi i, that's one,
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and we've come back again.
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00:11:56,450 --> 00:11:58,700
So it comes back to the start.
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00:11:58,700 --> 00:12:07,740
It's periodic, when this
square root of two i --
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00:12:07,740 --> 00:12:10,930
shall I call it capital
T, for the period?
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00:12:10,930 --> 00:12:17,270
For that particular T, if
that equals two pi i, then e
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to this thing is one, and
we've come around again.
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So the period is T is
determined here, cancel the i-s,
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and T is pi times the
square root of two.
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So that's pretty neat.
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00:12:32,270 --> 00:12:35,610
We get all the information
about all solutions,
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we haven't fixed on only
one particular solution,
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but it comes around again.
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So this was probably
my first chance
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00:12:43,730 --> 00:12:46,270
to say something
about the whole family
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00:12:46,270 --> 00:12:49,760
of anti-symmetric,
skew-symmetric matrices.
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OK.
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And then, finally, I asked,
take two eigenvectors (again,
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I haven't computed
the eigenvectors)
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and it turns out
they're orthogonal.
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They're orthogonal.
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The eigenvectors of
a symmetric matrix,
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or a skew-symmetric matrix,
are always orthogonal.
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I guess may conscience
makes me tell you,
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what are all the matrices that
have orthogonal eigenvectors?
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And symmetric is the
most important class,
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so that's the one
we've spoken about.
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But let me just put that
little fact down, here.
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Orthogonal x-s.
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00:13:36,661 --> 00:13:37,202
eigenvectors.
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00:13:41,430 --> 00:13:44,200
A matrix has orthogonal
eigenvectors,
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the exact condition -- it's
quite beautiful that I can tell
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you exactly when that happens.
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00:13:49,340 --> 00:13:55,290
It happens when A times A
transpose equals A transpose
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times A. Any time
that's the condition
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00:14:00,290 --> 00:14:03,700
for orthogonal eigenvectors.
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00:14:03,700 --> 00:14:09,040
And because we're interested
in special families of vectors,
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00:14:09,040 --> 00:14:12,690
tell me some special
families that fit.
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00:14:12,690 --> 00:14:15,410
This is the whole requirement.
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That's a pretty special
requirement most matrices have.
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00:14:21,120 --> 00:14:23,070
So the average
three-by-three matrix
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has three eigenvectors,
but not orthogonal.
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00:14:26,240 --> 00:14:29,380
But if it happens to
commute with its transpose,
256
00:14:29,380 --> 00:14:34,010
then, wonderfully, the
eigenvectors are orthogonal.
257
00:14:34,010 --> 00:14:39,250
Now, do you see how symmetric
matrices pass this test?
258
00:14:39,250 --> 00:14:40,340
Of course.
259
00:14:40,340 --> 00:14:43,790
If A transpose equals A, then
both sides are A squared,
260
00:14:43,790 --> 00:14:45,640
we've got it.
261
00:14:45,640 --> 00:14:49,410
How do anti-symmetric
matrices pass this test?
262
00:14:49,410 --> 00:14:54,060
If A transpose equals
minus A, then we've
263
00:14:54,060 --> 00:14:58,220
got it again, because we've got
minus A squared on both sides.
264
00:14:58,220 --> 00:15:00,050
So that's another group.
265
00:15:00,050 --> 00:15:03,090
And finally, let me ask you
about our other favorite
266
00:15:03,090 --> 00:15:06,950
family, orthogonal matrices.
267
00:15:06,950 --> 00:15:11,730
Do orthogonal matrices pass
this test, if A is a Q,
268
00:15:11,730 --> 00:15:14,990
do they pass the test for
orthogonal eigenvectors.
269
00:15:14,990 --> 00:15:22,320
Well, if A is Q, an orthogonal
matrix, what is Q transpose Q?
270
00:15:22,320 --> 00:15:23,020
It's I.
271
00:15:23,020 --> 00:15:25,310
And what is Q Q transpose?
272
00:15:25,310 --> 00:15:28,010
It's I, we're talking
square matrices here.
273
00:15:28,010 --> 00:15:30,100
So yes, it passes the test.
274
00:15:30,100 --> 00:15:36,600
So the special cases are
symmetric, anti-symmetric
275
00:15:36,600 --> 00:15:41,080
(I'll say skew-symmetric,)
and orthogonal.
276
00:15:41,080 --> 00:15:44,280
Those are the three
important special classes
277
00:15:44,280 --> 00:15:45,710
that are in this family.
278
00:15:45,710 --> 00:15:46,210
OK.
279
00:15:46,210 --> 00:15:52,860
That's like a comment that,
could have been made back in,
280
00:15:52,860 --> 00:15:54,870
section six point four.
281
00:15:54,870 --> 00:16:04,390
OK, I can pursue the
differential equations, also
282
00:16:04,390 --> 00:16:09,090
this question, didn't
ask you to tell me,
283
00:16:09,090 --> 00:16:13,920
how would I find this matrix
exponential, e to the At?
284
00:16:13,920 --> 00:16:15,050
So can I erase this?
285
00:16:15,050 --> 00:16:17,050
I'll just stay with this same...
286
00:16:19,690 --> 00:16:23,770
how would I find e to the At?
287
00:16:23,770 --> 00:16:27,010
Because, how does that come in?
288
00:16:27,010 --> 00:16:30,140
That's the key matrix for
a differential equation,
289
00:16:30,140 --> 00:16:32,460
because the solution is --
290
00:16:32,460 --> 00:16:38,520
the solution is
u(t) is e^(At) u(0).
291
00:16:38,520 --> 00:16:42,060
So this is like the
fundamental matrix
292
00:16:42,060 --> 00:16:48,400
that multiplies the given
function and gives the answer.
293
00:16:48,400 --> 00:16:53,630
And how would we compute
it if we wanted that?
294
00:16:53,630 --> 00:16:56,830
We don't always have to
find e to the At, because I
295
00:16:56,830 --> 00:17:00,500
can go directly to the answer
without any e to the At-s,
296
00:17:00,500 --> 00:17:06,440
but hiding here is an e to the
At, and how would I compute it?
297
00:17:06,440 --> 00:17:10,026
Well, if A is diagonalizable.
298
00:17:12,589 --> 00:17:21,510
So I'm now going to put in my
usual if A can be diagonalized
299
00:17:21,510 --> 00:17:25,240
(and everybody remember
that there is an if there,
300
00:17:25,240 --> 00:17:28,820
because it might not
have enough eigenvectors)
301
00:17:28,820 --> 00:17:33,330
this example does have enough,
random matrices have enough.
302
00:17:33,330 --> 00:17:36,720
So if we can diagonalize, then
we get a nice formula for this,
303
00:17:36,720 --> 00:17:40,050
because an S comes way
out at the beginning,
304
00:17:40,050 --> 00:17:42,740
and S inverse comes
way out at the end,
305
00:17:42,740 --> 00:17:47,450
and we only have to take
the exponential of lambda.
306
00:17:47,450 --> 00:17:49,980
And that's just a
diagonal matrix,
307
00:17:49,980 --> 00:17:53,780
so that's just e
the lambda one t,
308
00:17:53,780 --> 00:18:00,030
these guys are showing up,
now, in e to the lambda nt.
309
00:18:00,030 --> 00:18:01,070
OK?
310
00:18:01,070 --> 00:18:03,510
That's a really quick
review of that formula.
311
00:18:06,040 --> 00:18:08,780
It's something we can
compute it quickly
312
00:18:08,780 --> 00:18:11,250
if we have done the
S and lambda part.
313
00:18:13,770 --> 00:18:15,470
If we know S and
lambda, then it's
314
00:18:15,470 --> 00:18:17,440
not hard to take that step.
315
00:18:17,440 --> 00:18:20,770
OK, that's some comments
on differential equations.
316
00:18:20,770 --> 00:18:28,330
I would like to go on to a next
question that I started here.
317
00:18:28,330 --> 00:18:33,410
And it's, got several parts,
and I can just read it out.
318
00:18:33,410 --> 00:18:37,320
What we're given is a
three-by-three matrix,
319
00:18:37,320 --> 00:18:41,900
and we're told its eigenvalues,
except one of these
320
00:18:41,900 --> 00:18:47,700
is, like, we don't know, and
we're told the eigenvectors.
321
00:18:47,700 --> 00:18:50,500
And I want to ask
you about the matrix.
322
00:18:50,500 --> 00:18:51,190
OK.
323
00:18:51,190 --> 00:18:54,530
So, first question.
324
00:18:54,530 --> 00:18:56,430
Is the matrix diagonalizable?
325
00:18:59,000 --> 00:19:03,030
And I really mean for
which c, because I
326
00:19:03,030 --> 00:19:06,850
don't know c, so my
questions will all be,
327
00:19:06,850 --> 00:19:12,340
for which is there a condition
on c, does one c work.
328
00:19:12,340 --> 00:19:17,270
But your answer should tell
me all the c-s that work.
329
00:19:17,270 --> 00:19:21,390
I'm not asking for you to
tell me, well, c equal four,
330
00:19:21,390 --> 00:19:22,660
yes, that checks out.
331
00:19:22,660 --> 00:19:27,927
I want to know all the c-s
that make it diagonalizable.
332
00:19:34,950 --> 00:19:36,640
OK?
333
00:19:36,640 --> 00:19:39,440
What's the real
on diagonalizable?
334
00:19:39,440 --> 00:19:42,212
We need enough
eigenvectors, right?
335
00:19:42,212 --> 00:19:43,920
We don't care what
those eigenvalues are,
336
00:19:43,920 --> 00:19:46,710
it's eigenvectors that
count for diagonalizable,
337
00:19:46,710 --> 00:19:49,220
and we need three
independent ones,
338
00:19:49,220 --> 00:19:52,340
and are those three
guys independent?
339
00:19:52,340 --> 00:19:53,580
Yes.
340
00:19:53,580 --> 00:19:56,300
Actually, let's look
at them for a moment.
341
00:19:56,300 --> 00:19:59,921
What do you see about those
three vectors right away?
342
00:19:59,921 --> 00:20:01,170
They're more than independent.
343
00:20:04,380 --> 00:20:09,730
Can you see why those
three got chosen?
344
00:20:09,730 --> 00:20:15,780
Because it will come up in the
next part, they're orthogonal.
345
00:20:15,780 --> 00:20:17,920
Those eigenvectors
are orthogonal.
346
00:20:17,920 --> 00:20:19,710
They're certainly independent.
347
00:20:19,710 --> 00:20:28,220
So the answer to diagonalizable
is, yes, all c, all c.
348
00:20:28,220 --> 00:20:30,760
Doesn't matter. c could
be a repeated guy,
349
00:20:30,760 --> 00:20:32,360
but we've got
enough eigenvectors,
350
00:20:32,360 --> 00:20:33,930
so that's what we care about.
351
00:20:33,930 --> 00:20:36,400
OK, second question.
352
00:20:36,400 --> 00:20:38,365
For which values of
c is it symmetric?
353
00:20:40,960 --> 00:20:46,000
OK, what's the
answer to that one?
354
00:20:48,650 --> 00:20:53,420
If we know the same setup if
we know that much about it,
355
00:20:53,420 --> 00:20:55,330
we know those
eigenvectors, and we've
356
00:20:55,330 --> 00:21:02,850
noticed they're orthogonal,
then which c-s will work?
357
00:21:02,850 --> 00:21:07,800
So the eigenvalues of that
symmetric matrix have to be
358
00:21:07,800 --> 00:21:08,490
real.
359
00:21:08,490 --> 00:21:11,780
So all real c.
360
00:21:11,780 --> 00:21:17,300
If c was i, the matrix
wouldn't have been symmetric.
361
00:21:17,300 --> 00:21:24,040
But if c is a real number, then
we've got real eigenvalues,
362
00:21:24,040 --> 00:21:25,920
we've got orthogonal
eigenvectors,
363
00:21:25,920 --> 00:21:27,400
that matrix is symmetric.
364
00:21:27,400 --> 00:21:28,790
OK, positive definite.
365
00:21:28,790 --> 00:21:40,630
OK, now this is a
sub-case of symmetric,
366
00:21:40,630 --> 00:21:45,900
so we need c to be real, so
we've got a symmetric matrix,
367
00:21:45,900 --> 00:21:50,360
but we also want the thing
to be positive definite.
368
00:21:50,360 --> 00:21:52,340
Now, we're looking
at eigenvalues,
369
00:21:52,340 --> 00:21:54,740
we've got a lot of tests
for positive definite,
370
00:21:54,740 --> 00:21:57,250
but eigenvalues,
if we know them,
371
00:21:57,250 --> 00:22:01,100
is certainly a good,
quick, clean test.
372
00:22:01,100 --> 00:22:05,570
Could this matrix be
positive definite?
373
00:22:05,570 --> 00:22:06,640
No.
374
00:22:06,640 --> 00:22:10,180
No, because it's got
an eigenvalue zero.
375
00:22:10,180 --> 00:22:12,640
It could be positive
semi-definite,
376
00:22:12,640 --> 00:22:15,900
you know, like
consolation prize,
377
00:22:15,900 --> 00:22:19,320
if c was greater
or equal to zero,
378
00:22:19,320 --> 00:22:21,680
it would be positive
semi-definite.
379
00:22:21,680 --> 00:22:25,520
But it's not, no.
380
00:22:25,520 --> 00:22:30,670
Semi-definite, if I put that
comment in, semi-definite,
381
00:22:30,670 --> 00:22:35,010
that the condition would be
c greater or equal to zero.
382
00:22:35,010 --> 00:22:36,300
That would be all right.
383
00:22:36,300 --> 00:22:37,220
OK.
384
00:22:37,220 --> 00:22:38,330
Next part.
385
00:22:38,330 --> 00:22:39,675
Is it a Markov matrix?
386
00:22:44,260 --> 00:22:44,870
Hm.
387
00:22:44,870 --> 00:22:50,440
Could this matrix be, if I
choose the number c correctly,
388
00:22:50,440 --> 00:22:52,040
a Markov matrix?
389
00:22:58,750 --> 00:23:02,700
Well, what do we know
about Markov matrices?
390
00:23:02,700 --> 00:23:05,320
Mainly, we know something
about their eigenvalues.
391
00:23:05,320 --> 00:23:10,500
One eigenvalue is always one,
and the other eigenvalues
392
00:23:10,500 --> 00:23:13,430
are smaller.
393
00:23:13,430 --> 00:23:14,690
Not larger.
394
00:23:14,690 --> 00:23:17,380
So an eigenvalue
two can't happen.
395
00:23:17,380 --> 00:23:22,000
So the answer is, no, not a ma-
that's never a Markov matrix.
396
00:23:22,000 --> 00:23:22,750
OK?
397
00:23:22,750 --> 00:23:29,970
And finally, could one half
of A be a projection matrix?
398
00:23:29,970 --> 00:23:32,820
So could it- could this --
eh-eh could this be twice
399
00:23:32,820 --> 00:23:33,890
a projection matrix?
400
00:23:33,890 --> 00:23:35,840
So let me write it this way.
401
00:23:35,840 --> 00:23:39,530
Could A over two be
a projection matrix?
402
00:23:44,820 --> 00:23:46,820
OK, what are
projection matrices?
403
00:23:46,820 --> 00:23:48,650
They're real.
404
00:23:48,650 --> 00:23:53,160
I mean, th- they're symmetric,
so their eigenvalues are real.
405
00:23:53,160 --> 00:23:56,680
But more than that, we know what
those eigenvalues have to be.
406
00:23:56,680 --> 00:24:01,150
What do the eigenvalues of a
projection matrix have to be?
407
00:24:01,150 --> 00:24:05,670
See, that any nice
matrix we've got
408
00:24:05,670 --> 00:24:08,490
an idea about its eigenvalues.
409
00:24:08,490 --> 00:24:12,980
So the eigenvalues of
projection matrices are zero and
410
00:24:12,980 --> 00:24:13,970
one.
411
00:24:13,970 --> 00:24:16,710
Zero and one, only.
412
00:24:16,710 --> 00:24:21,510
Because P squared equals P,
let me call this matrix P,
413
00:24:21,510 --> 00:24:26,510
so P squared equals P, so
lambda squared equals lambda,
414
00:24:26,510 --> 00:24:30,570
because eigenvalues of P
squared are lambda squared,
415
00:24:30,570 --> 00:24:37,520
and we must have that, so
lambda equals zero or one.
416
00:24:37,520 --> 00:24:38,140
OK.
417
00:24:38,140 --> 00:24:42,060
Now what value of
c will work there?
418
00:24:42,060 --> 00:24:48,250
So, then, there are some
value that will work,
419
00:24:48,250 --> 00:24:50,300
and what will work?
420
00:24:50,300 --> 00:24:56,340
c equals zero will work,
or what else will work?
421
00:24:59,310 --> 00:25:02,690
c equal to two.
422
00:25:02,690 --> 00:25:06,260
Because if c is two, then
when we divide by two,
423
00:25:06,260 --> 00:25:09,880
this Eigenvalue of
two will drop to one,
424
00:25:09,880 --> 00:25:13,110
and so will the other one,
so, or c equal to two.
425
00:25:13,110 --> 00:25:15,640
OK, those are the
guys that will work,
426
00:25:15,640 --> 00:25:21,110
and it was the fact that those
eigenvectors were orthogonal,
427
00:25:21,110 --> 00:25:23,780
the fact that those
eigenvectors were orthogonal
428
00:25:23,780 --> 00:25:26,170
carried us a lot
of the way, here.
429
00:25:26,170 --> 00:25:29,400
If they weren't orthogonal, then
symmetric would have been dead,
430
00:25:29,400 --> 00:25:31,420
positive definite
would have been dead,
431
00:25:31,420 --> 00:25:33,150
projection would have been dead.
432
00:25:33,150 --> 00:25:37,090
But those eigenvectors
were orthogonal,
433
00:25:37,090 --> 00:25:40,180
so it came down to
the eigenvalues.
434
00:25:40,180 --> 00:25:45,270
OK, that was like a chance to
review a lot of this chapter.
435
00:25:50,790 --> 00:25:56,030
Shall I jump to the singular
value decomposition,
436
00:25:56,030 --> 00:26:04,140
then, as the third, topic
for, for the review?
437
00:26:04,140 --> 00:26:06,080
OK, so I'm going
to. jump to this.
438
00:26:06,080 --> 00:26:06,580
OK.
439
00:26:13,950 --> 00:26:16,990
So this is the singular
value decomposition,
440
00:26:16,990 --> 00:26:21,070
known to everybody as the SVD.
441
00:26:21,070 --> 00:26:27,720
And that's a factorization
of A into orthogonal times
442
00:26:27,720 --> 00:26:33,830
diagonal times orthogonal.
443
00:26:33,830 --> 00:26:41,030
And we always call those U
and sigma and V transpose.
444
00:26:41,030 --> 00:26:42,300
OK.
445
00:26:42,300 --> 00:26:46,660
And the key to that --
446
00:26:46,660 --> 00:26:51,170
this is for every
matrix, every A, every A.
447
00:26:51,170 --> 00:26:54,110
Rectangular, doesn't
matter, whatever,
448
00:26:54,110 --> 00:26:56,740
has this decomposition.
449
00:26:56,740 --> 00:26:59,070
So it's really important.
450
00:26:59,070 --> 00:27:04,930
And the key to it is to look
at things like A transpose A.
451
00:27:04,930 --> 00:27:07,360
Can we remember what
happens with A transpose A?
452
00:27:07,360 --> 00:27:11,300
If I just transpose that
I get V sigma transpose U
453
00:27:11,300 --> 00:27:15,420
transpose, that's
multiplying A, which is U,
454
00:27:15,420 --> 00:27:24,850
sigma V transpose, and the
result is V on the outside,
455
00:27:24,850 --> 00:27:27,990
s- U transpose U
is the identity,
456
00:27:27,990 --> 00:27:30,930
because it's an
orthogonal matrix.
457
00:27:30,930 --> 00:27:34,550
So I'm just left with
sigma transpose sigma
458
00:27:34,550 --> 00:27:39,340
in the middle, that's
a diagonal, possibly
459
00:27:39,340 --> 00:27:42,960
rectangular diagonal by its
transpose, so the result,
460
00:27:42,960 --> 00:27:46,036
this is orthogonal,
diagonal, orthogonal.
461
00:27:49,840 --> 00:27:55,620
So, I guess, actually, this
is the SVD for A transpose A.
462
00:27:55,620 --> 00:27:59,930
Here I see orthogonal,
diagonal, and orthogonal.
463
00:27:59,930 --> 00:28:00,440
Great.
464
00:28:00,440 --> 00:28:07,000
But a little more is happening.
465
00:28:07,000 --> 00:28:09,580
For A transpose
A, the difference
466
00:28:09,580 --> 00:28:13,690
is, the orthogonal
guys are the same.
467
00:28:13,690 --> 00:28:15,640
It's V and V transpose.
468
00:28:15,640 --> 00:28:17,290
What I seeing here?
469
00:28:17,290 --> 00:28:21,950
I'm seeing the factorization
for a symmetric matrix.
470
00:28:21,950 --> 00:28:23,220
This thing is symmetric.
471
00:28:26,790 --> 00:28:30,590
So in a symmetric case,
U is the same as V.
472
00:28:30,590 --> 00:28:33,210
U is the same as V for
this symmetric matrix,
473
00:28:33,210 --> 00:28:34,990
and, of course, we
see it happening.
474
00:28:34,990 --> 00:28:35,540
OK.
475
00:28:35,540 --> 00:28:39,760
So that tells us,
right away, what V is.
476
00:28:39,760 --> 00:28:49,650
V is the eigenvector
matrix for A transpose A.
477
00:28:49,650 --> 00:28:50,490
OK.
478
00:28:50,490 --> 00:28:57,070
Now, if you were here when I
lectured about this topic, when
479
00:28:57,070 --> 00:29:00,600
I gave the topic on singular
value decompositions,
480
00:29:00,600 --> 00:29:03,180
you'll remember that
I got into trouble.
481
00:29:06,150 --> 00:29:09,860
I'm sorry to remember that
myself, but it happened.
482
00:29:09,860 --> 00:29:10,550
OK.
483
00:29:10,550 --> 00:29:13,680
How did it happen?
484
00:29:13,680 --> 00:29:16,900
I was in great shape for
a while, cruising along.
485
00:29:16,900 --> 00:29:20,120
So I found the eigenvectors
for A transpose A.
486
00:29:20,120 --> 00:29:21,870
Good.
487
00:29:21,870 --> 00:29:24,800
I found the singular
values, what were they?
488
00:29:24,800 --> 00:29:26,520
What were the singular values?
489
00:29:26,520 --> 00:29:32,720
The singular value
number i, or --
490
00:29:32,720 --> 00:29:36,890
these are the guys in sigma --
491
00:29:36,890 --> 00:29:39,770
this is diagonal with
the number sigma in it.
492
00:29:39,770 --> 00:29:42,910
This diagonal is
sigma one, sigma two,
493
00:29:42,910 --> 00:29:46,090
up to the rank, sigma r,
those are the non-zero ones.
494
00:29:48,830 --> 00:29:51,100
So I found those,
and what are they?
495
00:29:51,100 --> 00:29:53,130
Remind me about that?
496
00:29:53,130 --> 00:29:59,630
Well, here, I'm seeing them
squared, so their squares are
497
00:29:59,630 --> 00:30:03,470
the eigenvalues
of A transpose A.
498
00:30:03,470 --> 00:30:04,760
Good.
499
00:30:04,760 --> 00:30:09,160
So I just take the square root,
if I want the eigenvalues of A
500
00:30:09,160 --> 00:30:10,020
transpose --
501
00:30:10,020 --> 00:30:11,990
If I want the sigmas
and I know these,
502
00:30:11,990 --> 00:30:14,540
I take the square root,
the positive square root.
503
00:30:14,540 --> 00:30:16,730
OK.
504
00:30:16,730 --> 00:30:20,610
Where did I run into trouble?
505
00:30:20,610 --> 00:30:25,220
Well, then, my final
step was to find U.
506
00:30:25,220 --> 00:30:28,270
And I didn't read the book.
507
00:30:28,270 --> 00:30:35,480
So, I did something that was
practically right, but --
508
00:30:35,480 --> 00:30:38,880
well, I guess practically
right is not quite the same.
509
00:30:38,880 --> 00:30:44,650
OK, so I thought, OK, I'll
look at A A transpose.
510
00:30:44,650 --> 00:30:47,420
What happened when I
looked at A A transpose?
511
00:30:47,420 --> 00:30:51,070
Let me just put it here,
and then I can feel it.
512
00:30:51,070 --> 00:30:53,620
OK, so here's A A transpose.
513
00:30:57,120 --> 00:31:01,050
So that's U sigma V
transpose, that's A,
514
00:31:01,050 --> 00:31:05,240
and then the transpose
is V sigma transpose,
515
00:31:05,240 --> 00:31:06,300
U sigma transpose.
516
00:31:06,300 --> 00:31:07,930
Fine.
517
00:31:07,930 --> 00:31:10,610
And then, in the middle
is the identity again,
518
00:31:10,610 --> 00:31:12,570
so it looks great.
519
00:31:12,570 --> 00:31:17,050
U sigma sigma
transpose, U transpose.
520
00:31:17,050 --> 00:31:18,760
Fine.
521
00:31:18,760 --> 00:31:26,570
All good, and now
these columns of U
522
00:31:26,570 --> 00:31:29,900
are the eigenvectors,
that's U is the eigenvector
523
00:31:29,900 --> 00:31:33,120
matrix for this guy.
524
00:31:33,120 --> 00:31:36,960
That was correct,
so I did that fine.
525
00:31:36,960 --> 00:31:38,600
Where did something go wrong?
526
00:31:38,600 --> 00:31:40,820
A sign went wrong.
527
00:31:40,820 --> 00:31:44,570
A sign went wrong because --
and now -- now I see, actually,
528
00:31:44,570 --> 00:31:49,140
somebody told me
right after class,
529
00:31:49,140 --> 00:31:53,910
we can't tell from this
description which sign to give
530
00:31:53,910 --> 00:31:55,200
the eigenvectors.
531
00:31:55,200 --> 00:32:00,570
If these are the
eigenvectors of this matrix,
532
00:32:00,570 --> 00:32:02,660
well, if you give
me an eigenvector
533
00:32:02,660 --> 00:32:04,790
and I change all
its signs, we've
534
00:32:04,790 --> 00:32:06,920
still got another eigenvector.
535
00:32:06,920 --> 00:32:08,970
So what I wasn't
able to determine
536
00:32:08,970 --> 00:32:13,940
(and I had a fifty-fifty
change and life let me down,)
537
00:32:13,940 --> 00:32:16,600
the signs I just
happened to pick
538
00:32:16,600 --> 00:32:19,070
for the eigenvectors,
one of them
539
00:32:19,070 --> 00:32:21,750
I should have reversed the sign.
540
00:32:21,750 --> 00:32:27,190
So, from this, I can't tell
whether the eigenvector
541
00:32:27,190 --> 00:32:31,150
or its negative is the
right one to use in there.
542
00:32:31,150 --> 00:32:34,750
So the right way to
do it is to, having
543
00:32:34,750 --> 00:32:38,640
settled on the
signs, the Vs also, I
544
00:32:38,640 --> 00:32:42,100
don't know which sign to
choose, but I choose one.
545
00:32:42,100 --> 00:32:43,220
I choose one.
546
00:32:43,220 --> 00:32:50,290
And then, instead,
I should have used
547
00:32:50,290 --> 00:32:53,950
the one that tells me what
sign to choose, the rule
548
00:32:53,950 --> 00:33:02,330
that A times a V is
sigma times the U.
549
00:33:02,330 --> 00:33:07,140
So, having decided on
the V, I multiply by A,
550
00:33:07,140 --> 00:33:09,640
I'll notice the factor
sigma coming out,
551
00:33:09,640 --> 00:33:11,520
and there will be a
unit vector there,
552
00:33:11,520 --> 00:33:17,310
and I now know
exactly what it is,
553
00:33:17,310 --> 00:33:20,380
and not only up to
a change of sign.
554
00:33:20,380 --> 00:33:22,390
So that's the good
and, of course,
555
00:33:22,390 --> 00:33:25,910
this is the main
point about the SVD.
556
00:33:25,910 --> 00:33:28,210
That's the point that
we've diagonalized,
557
00:33:28,210 --> 00:33:32,950
that's A times the
matrix of Vs equals
558
00:33:32,950 --> 00:33:37,710
U times the diagonal
matrix of sigmas.
559
00:33:37,710 --> 00:33:39,470
That's the same as that.
560
00:33:39,470 --> 00:33:39,970
OK.
561
00:33:39,970 --> 00:33:47,800
So that's, like,
correcting the wrong sign
562
00:33:47,800 --> 00:33:50,000
from that earlier lecture.
563
00:33:50,000 --> 00:33:52,810
And that would complete that,
so that's how you would compute
564
00:33:52,810 --> 00:33:54,040
the SVD.
565
00:33:54,040 --> 00:33:58,380
Now, on the quiz, I going to
ask -- well, maybe on the final.
566
00:33:58,380 --> 00:34:01,010
So we've got quiz
and final ahead.
567
00:34:01,010 --> 00:34:05,400
Sometimes, you might be asked
to find the SVD if I give you
568
00:34:05,400 --> 00:34:10,870
the matrix -- let me come
back, now, to the main board --
569
00:34:10,870 --> 00:34:17,880
or, I might give you the pieces.
570
00:34:17,880 --> 00:34:21,810
And I might ask you
something about the matrix.
571
00:34:21,810 --> 00:34:31,580
For example, suppose I
ask you, oh, let's say,
572
00:34:31,580 --> 00:34:36,590
if I tell you what sigma is --
573
00:34:36,590 --> 00:34:37,460
OK.
574
00:34:37,460 --> 00:34:39,230
Let's take one example.
575
00:34:39,230 --> 00:34:43,820
Suppose sigma is --
576
00:34:43,820 --> 00:34:46,350
so all that's how we
would compute them.
577
00:34:46,350 --> 00:34:48,070
But now, suppose
I give you these.
578
00:34:48,070 --> 00:34:52,320
Suppose I give you sigma
is, say, three two.
579
00:34:57,130 --> 00:35:02,110
And I tell you that U
has a couple of columns,
580
00:35:02,110 --> 00:35:04,580
and V has a couple of columns.
581
00:35:07,910 --> 00:35:10,330
OK.
582
00:35:10,330 --> 00:35:12,600
Those are orthogonal
columns, of course,
583
00:35:12,600 --> 00:35:14,630
because U and V are orthogonal.
584
00:35:14,630 --> 00:35:16,120
I'm just sort of,
like, getting you
585
00:35:16,120 --> 00:35:19,440
to think about the SVD,
because we only had that one
586
00:35:19,440 --> 00:35:22,570
lecture about it,
and one homework,
587
00:35:22,570 --> 00:35:28,460
and, what kind of a
matrix have I got here?
588
00:35:28,460 --> 00:35:31,970
What do I know
about this matrix?
589
00:35:31,970 --> 00:35:35,540
All I really know right now
is that its singular values,
590
00:35:35,540 --> 00:35:39,390
those sigmas are three and
two, and the only thing
591
00:35:39,390 --> 00:35:43,190
interesting that I can see in
that is that they're not zero.
592
00:35:43,190 --> 00:35:48,390
I know that this matrix
is non-singular, right?
593
00:35:48,390 --> 00:35:51,570
That's invertible, I don't
have any zero eigenvalues,
594
00:35:51,570 --> 00:35:54,710
and zero singular values,
that's invertible,
595
00:35:54,710 --> 00:36:02,290
there's a typical SVD for a
nice two-by-two non-singular
596
00:36:02,290 --> 00:36:04,990
invertible good matrix.
597
00:36:04,990 --> 00:36:07,480
If I actually gave you
a matrix, then you'd
598
00:36:07,480 --> 00:36:10,390
have to find the Us and
the Vs as we just spoke.
599
00:36:10,390 --> 00:36:12,000
But, there.
600
00:36:12,000 --> 00:36:16,400
Now, what if the two
wasn't a two but it was --
601
00:36:16,400 --> 00:36:18,520
well, let me make an
extreme case, here --
602
00:36:18,520 --> 00:36:20,050
suppose it was minus five.
603
00:36:23,220 --> 00:36:24,810
That's wrong, right away.
604
00:36:24,810 --> 00:36:28,590
That's not a singular
value decomposition, right?
605
00:36:28,590 --> 00:36:30,840
The singular values
are not negative.
606
00:36:30,840 --> 00:36:36,011
So that's not a singular value
decomposition, and forget it.
607
00:36:36,011 --> 00:36:36,510
OK.
608
00:36:36,510 --> 00:36:40,200
So let me ask you
about that one.
609
00:36:40,200 --> 00:36:42,095
What can you tell me
about that matrix?
610
00:36:45,090 --> 00:36:47,340
It's singular, right?
611
00:36:47,340 --> 00:36:50,480
It's got a singular matrix
there in the middle,
612
00:36:50,480 --> 00:36:56,110
and, let's see, so,
OK, it's singular,
613
00:36:56,110 --> 00:37:01,870
maybe you can tell me, its rank?
614
00:37:01,870 --> 00:37:04,320
What's the rank of A?
615
00:37:04,320 --> 00:37:08,900
It's clearly --
somebody just say it --
616
00:37:08,900 --> 00:37:09,920
one, thanks.
617
00:37:09,920 --> 00:37:15,160
The rank is one,
so the null space,
618
00:37:15,160 --> 00:37:18,850
what's the dimension
of the null space?
619
00:37:18,850 --> 00:37:19,890
One.
620
00:37:19,890 --> 00:37:20,390
Right?
621
00:37:20,390 --> 00:37:23,940
We've got a two-by-two
matrix of rank one,
622
00:37:23,940 --> 00:37:26,820
so of all that stuff from
the beginning of the course
623
00:37:26,820 --> 00:37:30,310
is still with us.
624
00:37:30,310 --> 00:37:32,790
The dimensions of those
fundamental spaces
625
00:37:32,790 --> 00:37:36,950
is still central,
and a basis for them.
626
00:37:36,950 --> 00:37:40,890
Now, can you tell me a vector
that's in the null space?
627
00:37:40,890 --> 00:37:47,000
And then that will be my last
point to make about the SVD.
628
00:37:47,000 --> 00:37:49,400
Can you tell me a vector
that's in the null space?
629
00:37:54,410 --> 00:38:00,820
So what would I multiply
by and get zero, here?
630
00:38:00,820 --> 00:38:04,120
I think the answer
is probably v2.
631
00:38:04,120 --> 00:38:07,820
I think probably v2
is in the null space,
632
00:38:07,820 --> 00:38:11,950
because I think that must
be the eigenvector going
633
00:38:11,950 --> 00:38:14,800
with this zero eigenvalue.
634
00:38:14,800 --> 00:38:16,310
Yes.
635
00:38:16,310 --> 00:38:17,200
Have a look at that.
636
00:38:17,200 --> 00:38:21,390
And I could ask you the
null space of A transpose.
637
00:38:21,390 --> 00:38:23,050
And I could ask you
the column space.
638
00:38:23,050 --> 00:38:24,620
All that stuff.
639
00:38:24,620 --> 00:38:27,180
Everything is sitting
there in the SVD.
640
00:38:27,180 --> 00:38:29,990
The SVD takes a little
more time to compute,
641
00:38:29,990 --> 00:38:36,090
but it displays all the
good stuff about a matrix.
642
00:38:36,090 --> 00:38:36,720
OK.
643
00:38:36,720 --> 00:38:39,680
Any question about the SVD?
644
00:38:39,680 --> 00:38:47,800
Let me keep going
with further topics.
645
00:38:47,800 --> 00:38:48,990
Now, let's see.
646
00:38:48,990 --> 00:38:51,050
Similar matrices
we've talked about,
647
00:38:51,050 --> 00:38:55,390
let me see if I've
got another, --
648
00:38:55,390 --> 00:38:57,620
OK.
649
00:38:57,620 --> 00:39:04,570
Here's a true false, so
we can do that, easily.
650
00:39:04,570 --> 00:39:05,300
So.
651
00:39:05,300 --> 00:39:07,560
Question, A given.
652
00:39:10,480 --> 00:39:17,840
A is symmetric and orthogonal.
653
00:39:20,861 --> 00:39:21,360
OK.
654
00:39:26,920 --> 00:39:29,870
So beautiful matrices like that
don't come along every day.
655
00:39:29,870 --> 00:39:36,480
But what can we say first
about its eigenvalues?
656
00:39:36,480 --> 00:39:38,680
Actually, of course.
657
00:39:38,680 --> 00:39:41,810
Here are our two most
important classes of matrices,
658
00:39:41,810 --> 00:39:45,500
and we're looking
at the intersection.
659
00:39:45,500 --> 00:39:48,470
So those really
are neat matrices,
660
00:39:48,470 --> 00:39:50,920
and what can you tell
me about what could
661
00:39:50,920 --> 00:39:52,670
the possible eigenvalues be?
662
00:39:52,670 --> 00:39:57,540
Eigenvalues can be what?
663
00:39:57,540 --> 00:39:59,330
What do I know about
the eigenvalues
664
00:39:59,330 --> 00:40:01,280
of a symmetric matrix?
665
00:40:01,280 --> 00:40:04,690
Lambda is real.
666
00:40:04,690 --> 00:40:06,550
What do I know about
the eigenvalues
667
00:40:06,550 --> 00:40:10,260
of an orthogonal matrix?
668
00:40:10,260 --> 00:40:12,370
Ha.
669
00:40:12,370 --> 00:40:13,200
Maybe nothing.
670
00:40:13,200 --> 00:40:15,420
But, no, that can't be.
671
00:40:15,420 --> 00:40:17,911
What do I know about the
eigenvalues of an orthogonal
672
00:40:17,911 --> 00:40:18,410
matrix?
673
00:40:18,410 --> 00:40:19,565
Well, what feels right?
674
00:40:22,330 --> 00:40:26,620
Basing mathematics on just
a little gut instinct here,
675
00:40:26,620 --> 00:40:29,910
the eigenvalues of
an orthogonal matrix
676
00:40:29,910 --> 00:40:33,800
ought to have magnitude one.
677
00:40:33,800 --> 00:40:36,280
Orthogonal matrices
are like rotations,
678
00:40:36,280 --> 00:40:40,921
they're not changing the length,
so orthogonal, the eigenvalues
679
00:40:40,921 --> 00:40:41,420
are one.
680
00:40:41,420 --> 00:40:50,070
Let me just show you why.
681
00:40:50,070 --> 00:40:53,920
So the matrix, can I
call it Q for orthogonal
682
00:40:53,920 --> 00:40:55,810
Why? for the moment?
683
00:40:55,810 --> 00:40:58,760
If I look at Q x
equal lambda x, how
684
00:40:58,760 --> 00:41:03,570
do I see that this
thing has magnitude one?
685
00:41:03,570 --> 00:41:06,140
I take the length of both sides.
686
00:41:06,140 --> 00:41:08,930
This is taking lengths,
taking lengths,
687
00:41:08,930 --> 00:41:13,520
this is whatever the magnitude
is times the length of x.
688
00:41:13,520 --> 00:41:18,170
And what's the length of Q x
if Q is an orthogonal matrix?
689
00:41:18,170 --> 00:41:20,910
This is something
you should know.
690
00:41:20,910 --> 00:41:23,100
It's the same as
the length of x.
691
00:41:23,100 --> 00:41:26,400
Orthogonal matrices
don't change lengths.
692
00:41:26,400 --> 00:41:30,330
So lambda has to be one.
693
00:41:30,330 --> 00:41:31,090
Right.
694
00:41:31,090 --> 00:41:31,890
OK.
695
00:41:31,890 --> 00:41:34,440
That's worth
committing to memory,
696
00:41:34,440 --> 00:41:36,930
that could show up again.
697
00:41:36,930 --> 00:41:37,430
OK.
698
00:41:37,430 --> 00:41:40,720
So what's the answer
now to this question,
699
00:41:40,720 --> 00:41:42,940
what can the eigenvalues be?
700
00:41:42,940 --> 00:41:45,220
There's only two
possibilities, and they
701
00:41:45,220 --> 00:41:55,050
are one and the other
one, the other possibility
702
00:41:55,050 --> 00:42:00,770
is negative one, right, because
these have the right magnitude,
703
00:42:00,770 --> 00:42:03,090
and they're real.
704
00:42:03,090 --> 00:42:04,070
OK.
705
00:42:04,070 --> 00:42:04,730
TK.
706
00:42:04,730 --> 00:42:07,090
true -- OK.
707
00:42:07,090 --> 00:42:08,260
True or false?
708
00:42:08,260 --> 00:42:12,220
A is sure to be
positive definite.
709
00:42:12,220 --> 00:42:14,070
Well, this is a great
matrix, but is it
710
00:42:14,070 --> 00:42:16,570
sure to be positive definite?
711
00:42:16,570 --> 00:42:17,090
No.
712
00:42:17,090 --> 00:42:19,380
If it could have an
eigenvalue minus one,
713
00:42:19,380 --> 00:42:21,910
it wouldn't be
positive definite.
714
00:42:21,910 --> 00:42:24,230
True or false, it has
no repeated eigenvalues.
715
00:42:27,820 --> 00:42:29,700
That's false, too.
716
00:42:29,700 --> 00:42:31,710
In fact, it's going to
have repeated eigenvalues
717
00:42:31,710 --> 00:42:33,740
if it's as big as
three by three,
718
00:42:33,740 --> 00:42:35,630
one of these c- one
of these, at least,
719
00:42:35,630 --> 00:42:37,270
will have to get repeated.
720
00:42:37,270 --> 00:42:37,770
Sure.
721
00:42:37,770 --> 00:42:40,040
So it's got repeated
eigenvalues, but,
722
00:42:40,040 --> 00:42:42,910
is it diagonalizable?
723
00:42:42,910 --> 00:42:45,210
It's got these many, many,
repeated eigenvalues.
724
00:42:45,210 --> 00:42:46,900
If it's fifty by
fifty, it's certainly
725
00:42:46,900 --> 00:42:48,790
got a lot of repetitions.
726
00:42:48,790 --> 00:42:51,370
Is it diagonalizable?
727
00:42:51,370 --> 00:42:52,110
Yes.
728
00:42:52,110 --> 00:42:55,500
All symmetric matrices,
all orthogonal matrices
729
00:42:55,500 --> 00:42:57,090
can be diagonalized.
730
00:42:57,090 --> 00:43:02,620
And, in fact, the eigenvectors
can even be chosen orthogonal.
731
00:43:02,620 --> 00:43:05,120
So it could be, sort
of, like, diagonalized
732
00:43:05,120 --> 00:43:09,270
the best way with a Q,
and not just any old S.
733
00:43:09,270 --> 00:43:09,840
OK.
734
00:43:09,840 --> 00:43:11,820
Is it non-singular?
735
00:43:11,820 --> 00:43:15,275
Is a symmetric orthogonal
matrix non-singular?
736
00:43:19,500 --> 00:43:21,790
Orthogonal matrices are
always non-singular.
737
00:43:21,790 --> 00:43:22,900
Sure.
738
00:43:22,900 --> 00:43:26,910
And, obviously, we don't
have any zero Eigenvalues.
739
00:43:26,910 --> 00:43:28,670
Is it sure to be diagonalizable?
740
00:43:28,670 --> 00:43:31,310
Yes.
741
00:43:31,310 --> 00:43:41,370
Now, here's a final step -- show
that one-half of A plus I is A
742
00:43:41,370 --> 00:43:42,140
--
743
00:43:42,140 --> 00:43:51,705
that is, prove one-half of A
plus I is a projection matrix.
744
00:43:58,880 --> 00:43:59,380
OK?
745
00:44:04,080 --> 00:44:04,750
Let's see.
746
00:44:04,750 --> 00:44:05,510
What do I do?
747
00:44:09,170 --> 00:44:11,610
I could see two ways to do this.
748
00:44:11,610 --> 00:44:14,560
I could check the properties
of a projection matrix, which
749
00:44:14,560 --> 00:44:15,700
are what?
750
00:44:15,700 --> 00:44:18,520
A projection matrix
is symmetric.
751
00:44:18,520 --> 00:44:21,790
Well, that's certainly
symmetric, because A is.
752
00:44:21,790 --> 00:44:24,440
And what's the other property?
753
00:44:24,440 --> 00:44:26,230
I should square
it, and hopefully
754
00:44:26,230 --> 00:44:27,900
get the same thing back.
755
00:44:27,900 --> 00:44:32,230
So can I do that, square and see
if I get the same thing back?
756
00:44:32,230 --> 00:44:37,060
So if I square it, I'll get
one-quarter of A squared
757
00:44:37,060 --> 00:44:40,640
plus two A plus I, right?
758
00:44:40,640 --> 00:44:48,430
And the question is, does that
agree with p- the thing itself?
759
00:44:48,430 --> 00:44:53,480
One-half A plus I.
760
00:44:53,480 --> 00:44:53,980
Hm.
761
00:44:57,240 --> 00:45:02,060
I guess I'd like to know
something about A squared.
762
00:45:02,060 --> 00:45:03,260
What is A squared?
763
00:45:03,260 --> 00:45:05,090
That's our problem.
764
00:45:05,090 --> 00:45:06,260
What is A squared?
765
00:45:11,560 --> 00:45:13,510
If A is symmetric
and orthogonal,
766
00:45:13,510 --> 00:45:17,390
A is symmetric and orthogonal.
767
00:45:22,060 --> 00:45:23,710
This is what we're given, right?
768
00:45:23,710 --> 00:45:27,990
It's symmetric, and
it's orthogonal.
769
00:45:27,990 --> 00:45:31,350
So what's A squared?
770
00:45:31,350 --> 00:45:36,360
I. A squared is I,
because A times A --
771
00:45:36,360 --> 00:45:42,500
if A equals its own inverse,
so A times A is the same as A
772
00:45:42,500 --> 00:45:46,150
times A inverse, which is I.
773
00:45:46,150 --> 00:45:52,320
So this A squared here is I.
774
00:45:52,320 --> 00:45:54,530
And now we've got it.
775
00:45:54,530 --> 00:45:57,810
We've got two identities
over four, that's good,
776
00:45:57,810 --> 00:46:01,060
and we've got two As
over four, that's good.
777
00:46:01,060 --> 00:46:01,670
OK.
778
00:46:01,670 --> 00:46:05,960
So it turned out to be a
projection matrix safely.
779
00:46:05,960 --> 00:46:08,310
And we could also
have said, well,
780
00:46:08,310 --> 00:46:11,230
what are the eigenvalues
of this thing?
781
00:46:11,230 --> 00:46:14,870
What are the eigenvalues
of a half A plus I?
782
00:46:14,870 --> 00:46:17,970
If the eigenvalues of A
are one and minus one,
783
00:46:17,970 --> 00:46:21,840
what are the
eigenvalues of A plus I?
784
00:46:21,840 --> 00:46:25,860
Just stay with it these
last thirty seconds here.
785
00:46:25,860 --> 00:46:29,140
What if I know these
eigenvalues of A,
786
00:46:29,140 --> 00:46:31,480
and I add the identity,
the eigenvalues
787
00:46:31,480 --> 00:46:35,600
of A plus I are zero and two.
788
00:46:35,600 --> 00:46:39,480
And then when I divide by two,
the eigenvalues are zero and
789
00:46:39,480 --> 00:46:40,020
one.
790
00:46:40,020 --> 00:46:43,130
So it's symmetric, it's
got the right eigenvalues,
791
00:46:43,130 --> 00:46:45,070
it's a projection matrix.
792
00:46:45,070 --> 00:46:49,290
OK, you're seeing a lot of
stuff about eigenvalues,
793
00:46:49,290 --> 00:46:54,460
and special matrices, and
that's what the quiz is about.
794
00:46:54,460 --> 00:46:57,230
OK, so good luck on the quiz.