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Okay.
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This lecture is mostly about
the idea of similar matrixes.
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I'm going to tell you what
that word similar means
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and in what way two
matrixes are called similar.
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But before I do that,
I have a little more
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to say about positive
definite matrixes.
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You can tell this is a subject I
think is really important and I
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told you what positive
definite meant --
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it means that this --
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this expression, this
quadratic form, x transpose I
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x is always positive.
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But the direct way to test
it was with eigenvalues
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or pivots or determinants.
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So I -- we know what it
means, we know how to test it,
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but I didn't really say where
positive definite matrixes come
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from.
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And so one thing I want to say
is that they come from least
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squares in -- and all sorts of
physical problems start with
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a rectangular matrix -- well,
you remember in least squares
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the crucial combination
was A transpose A.
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So I want to show that that's
a positive definite matrix.
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Can -- so I --
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I'm going to speak a little
more about positive definite
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matrixes, just recapping --
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so let me ask a question.
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It may be on the homework.
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Suppose a matrix A
is positive definite.
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I mean by that it's all --
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I'm assuming it's symmetric.
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That's always built
into the definition.
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So we have a symmetric
positive definite matrix.
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What about its inverse?
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Is the inverse of a symmetric
positive definite matrix also
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symmetric positive definite?
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So you quickly think,
okay, what do I
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know about the pivots
of the inverse matrix?
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Not much.
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What do I know about
the eigenvalues
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of the inverse matrix?
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Everything, right?
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The eigenvalues
of the inverse are
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one over the eigenvalues
of the matrix.
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So if my matrix starts
out positive definite,
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then right away I know that its
inverse is positive definite,
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because those positive
eigenvalues --
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then one over the
eigenvalue is also positive.
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What if I know that A -- a
matrix A and a matrix B are
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both positive definite?
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But let me ask you this.
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Suppose if A and B are positive
definite, what about --
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what about A plus B?
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In some way, you hope
that that would be true.
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It's -- positive definite for a
matrix is kind of like positive
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for a real number.
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But we don't know the
eigenvalues of A plus B.
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We don't know the
pivots of A plus B.
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So we just, like, have to go
down this list of, all right,
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which approach to
positive definite
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can we get a handle on?
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And this is a good one.
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This is a good one.
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Can we -- how would
we decide that --
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if A was like this and
if B was like this,
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then we would look at
x transpose A plus B x.
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I'm sure this is
in the homework.
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Now -- so we have x transpose
A x bigger than zero,
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x transpose B x positive
for all -- for all x,
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so now I ask you about this
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guy.
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And of course, you
just add that and that
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and we get what we want.
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If A and B are positive
definites, so is A plus B.
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So that's what I've shown.
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So is A plus B.
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Just -- be sort of ready for
all the approaches through
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eigenvalues and through
this expression.
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And now, finally, one more
thought about positive definite
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is this combination that
came up in least squares.
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Can I do that?
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So now -- now suppose A
is rectangular, m by n.
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I -- so I'm sorry that
I've used the same letter A
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for the positive definite
matrixes in the eigenvalue
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chapter that I used way back
in earlier chapters when
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the matrix was rectangular.
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Now, that matrix --
a rectangular matrix,
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no way its positive definite.
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It's not symmetric.
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It's not even square in general.
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But you remember that the key
for these rectangular ones
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was A transpose A.
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That's square.
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That's symmetric.
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Those are things we knew --
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we knew back when
we met this thing
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in the least square stuff,
in the projection stuff.
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But now we know
something more --
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we can ask a more important
question, a deeper question --
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is it positive definite?
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And we sort of hope so.
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Like, we -- we might --
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in analogy with
numbers, this is like --
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sort of like the square of a
number, and that's positive.
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So now I want to ask
the matrix question.
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Is A transpose A
positive definite?
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Okay, now it's -- so again,
it's a rectangular A that
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I'm starting with, but it's
the combination A transpose A
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that's the square, symmetric
and hopefully positive definite
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matrix.
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So how -- how do I see that
it is positive definite,
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or at least positive
semi-definite?
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You'll see that.
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Well, I don't know the
eigenvalues of this product.
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I don't want to work
with the pivots.
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The right thing -- the right
quantity to look at is this,
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x transpose Ax --
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A -- x transpose times
my matrix times x.
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I'd like to see
that this thing --
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that that expression
is always positive.
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I'm not doing it with numbers,
I'm doing it with symbols.
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Do you see -- how do I see
that that expression comes out
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positive?
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I'm taking a rectangular
matrix A and an A transpose --
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that gives me something
square symmetric,
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but now I want to see
that if I multiply --
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that if I do this --
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I form this quadratic
expression that I
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get this positive thing that
goes upwards when I graph it.
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How do I see that
that's positive,
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or absolutely it
isn't negative anyway?
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We'll have to, like, spend
a minute on the question
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could it be zero, but
it can't be negative.
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Why can this never be negative?
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The argument is --
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like the one key idea in so
many steps in linear algebra --
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put those parentheses
in a good way.
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Put the parentheses around
Ax and what's the first part?
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What's this x
transpose A transpose?
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That is Ax transpose.
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So what do we have?
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We have the length
squared of Ax.
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We have -- that's the column
vector Ax that's the row vector
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Ax, its length squared,
certainly greater than
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or possibly equal to zero.
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So we have to deal with
this little possibility.
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Could it be equal?
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Well, when could the
length squared be zero?
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Only if the vector
is zero, right?
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That's the only vector that
has length squared zero.
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So we have -- we
would like to --
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I would like to get that
possibility out of there.
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So I want to have Ax
never -- never be zero,
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except of course
for the zero vector.
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How do I assure that
Ax is never zero?
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The -- in other words, how do I
show that there's no null space
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of A?
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The rank should be --
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so now remember -- what's
the rank when there's no null
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space?
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By no null space,
you know what I mean.
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Only the zero vector
in the null space.
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So if I have a -- if I
have an 11 by 5 matrix --
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so it's got 11 rows, 5 columns,
when is there no null space?
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So the columns should be
independent -- what's the rank?
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n 5 -- rank n.
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Independent columns,
when -- so if I --
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then I conclude yes,
positive definite.
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And this was the assumption
-- then A transpose A is
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invertible --
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the least squares
equations all work fine.
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And more than that -- the matrix
is even positive definite.
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And I just to say one comment
about numerical things,
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with a positive definite
matrix, you never
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have to do row exchanges.
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You never run into unsuitably
small numbers or zeroes
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in the pivot position.
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They're the right -- they're the
great matrixes to compute with,
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and they're the great
matrixes to study.
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So that's -- I wanted to take
this first ten minutes of grab
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the first ten minutes away from
similar matrixes and continue
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a -- this much more
with positive definite.
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I'm really at this
point, now, coming close
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to the end of the heart
of linear algebra.
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The positive definiteness
brought everything together.
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Similar matrixes, which is
coming the rest of this hour
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is a key topic, and
please come on Monday.
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Monday is about what's called
the SVD, singular values.
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It's the -- has become
a central fact in --
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a central part of
linear algebra.
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I mean, you can come
after Monday also, but --
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Monday is, -- that singular
value thing has made it
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into this course.
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Ten years ago, five years
ago it wasn't in the course,
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now it has to be.
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Okay.
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So can I begin today's
lecture proper with this idea
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of similar matrixes.
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This is what similar
matrixes mean.
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So here -- let's start again.
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I'll write it again.
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So A and B are similar.
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A and B are -- now I'm
-- these matrixes --
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I'm no longer talking about
symmetric matrixes, in --
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at least no longer expecting
symmetric matrixes.
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I'm talking about two
square matrixes n by n.
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A and B, they're
n by n matrixes.
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And I'm introducing
this word similar.
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So I'm going to say
what does it mean?
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It means that they're
connected in the way --
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well, in the way I've written
here, so let me rewrite it.
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That means that for some matrix
M, which has to be invertible,
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because you'll see that --
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this one matrix is --
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00:13:29,680 --> 00:13:32,980
take the other matrix,
multiply on the right
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00:13:32,980 --> 00:13:39,700
by M and on the
left by M inverse.
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00:13:39,700 --> 00:13:43,810
So the question is,
why that combination?
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But part of the answer
you know already.
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You remember -- we've done this
-- we've taken a matrix A --
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00:13:52,070 --> 00:13:58,620
so let's do an
example of similar.
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Suppose A -- the matrix A
-- suppose it has a full set
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of eigenvectors.
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They go in this
eigenvector matrix S.
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00:14:12,910 --> 00:14:15,220
Then what was the main
point of the whole --
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00:14:15,220 --> 00:14:19,960
the main calculation of the
whole chapter was -- is --
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00:14:19,960 --> 00:14:25,380
use that eigenvector
matrix S and its inverse
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00:14:25,380 --> 00:14:33,720
comes over there to produce the
nicest possible matrix lambda.
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Nicest possible
because it's diagonal.
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So in our new language, this is
saying A is similar to lambda.
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A is similar to lambda,
because there is a matrix,
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and this particular --
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there is an M and
this particular M
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is this important guy,
this eigenvector matrix.
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But if I take a different matrix
M and I look at M inverse A M,
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the result won't
come out diagonal,
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but it will come out a
matrix B that's similar to A.
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Do you see that I'm --
what I'm doing is, like --
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I'm putting these
matrixes into families.
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All the matrixes in one -- in
the family are similar to each
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other.
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They're all -- each one in this
family is connected to each
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other one by some
matrix M and the --
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like the outstanding member of
the family is the diagonal guy.
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I mean, that's the
simplest, neatest matrix
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in this family of all the
matrixes that are similar to A,
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the best one is lambda.
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But there are lots of others,
because I can take different --
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instead of S, I can
take any old matrix M,
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any old invertible
matrix and -- and do it.
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I'd better do an example.
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Okay.
250
00:16:10,520 --> 00:16:16,901
Suppose I take A as the
matrix two one one two.
251
00:16:16,901 --> 00:16:17,400
Okay.
252
00:16:21,510 --> 00:16:25,470
Do you know the eigenvalue
matrix for that?
253
00:16:25,470 --> 00:16:28,630
The eigenvalues of
that matrix are --
254
00:16:28,630 --> 00:16:33,040
well, three and one.
255
00:16:33,040 --> 00:16:37,430
So that -- and the eigenvectors
would be easy to find.
256
00:16:37,430 --> 00:16:40,780
So this matrix is
similar to this one.
257
00:16:40,780 --> 00:16:43,080
But my point is --
258
00:16:43,080 --> 00:16:49,410
but also, I can also take
my matrix, two one one two,
259
00:16:49,410 --> 00:16:52,090
I could multiply it by
-- let's see, what --
260
00:16:52,090 --> 00:16:55,580
I'm just going to cook
up a matrix M here.
261
00:16:55,580 --> 00:17:00,350
I'm -- I'll -- let me just
invent -- one four one zero.
262
00:17:00,350 --> 00:17:02,710
And over here I'll
put M inverse,
263
00:17:02,710 --> 00:17:05,839
and because I happened
to make that triangular,
264
00:17:05,839 --> 00:17:09,940
I know that its
inverse is that, right?
265
00:17:09,940 --> 00:17:13,819
So there's M inverse A M, that's
going to produce some matrix --
266
00:17:13,819 --> 00:17:17,200
oh, well, I've got to
do the multiplication,
267
00:17:17,200 --> 00:17:19,380
so hang on a second, let --
268
00:17:19,380 --> 00:17:22,880
I'll just copy that
one minus four zero one
269
00:17:22,880 --> 00:17:33,930
and multiply these guys so I'm
getting two nine one and six,
270
00:17:33,930 --> 00:17:36,010
I think.
271
00:17:36,010 --> 00:17:39,680
Can you check it as I go,
because you -- see I'm just --
272
00:17:39,680 --> 00:17:43,850
so that's two minus four,
I'm getting a minus two nine
273
00:17:43,850 --> 00:17:48,550
minus 24 is a minus 15, my
God, how did I get this?
274
00:17:48,550 --> 00:17:50,935
And that's probably one and six.
275
00:17:54,560 --> 00:17:57,810
So there's my matrix B.
276
00:17:57,810 --> 00:18:02,180
And there's my matrix
lambda, there's my matrix A
277
00:18:02,180 --> 00:18:04,715
and my point is these
are all similar matrixes.
278
00:18:07,250 --> 00:18:09,880
They all have
something in common,
279
00:18:09,880 --> 00:18:13,130
besides being just two by two.
280
00:18:13,130 --> 00:18:17,190
They have something in common.
281
00:18:17,190 --> 00:18:21,310
And that's -- and what is it?
282
00:18:21,310 --> 00:18:26,540
What's the point about two
matrixes that are built out
283
00:18:26,540 --> 00:18:27,840
of --
284
00:18:27,840 --> 00:18:32,440
the B is built out
of M inverse A M.
285
00:18:32,440 --> 00:18:34,840
What is it that A
and B have in common?
286
00:18:34,840 --> 00:18:38,127
That's the main -- now I'm
telling you the main fact about
287
00:18:38,127 --> 00:18:38,835
similar matrixes.
288
00:18:41,590 --> 00:18:44,910
They have the same eigenvalues.
289
00:18:44,910 --> 00:18:47,550
This is -- this chapter
is about eigenvalues,
290
00:18:47,550 --> 00:18:51,350
and that's why we're interested
in this family of matrixes that
291
00:18:51,350 --> 00:18:53,140
have the same eigenvalues.
292
00:18:53,140 --> 00:18:56,640
What are the eigenvalues
in this example?
293
00:18:56,640 --> 00:18:57,570
Lambda.
294
00:18:57,570 --> 00:19:01,070
The eigenvalues of
that I could compute.
295
00:19:01,070 --> 00:19:06,370
The eigenvalues of that I
can compute really fast.
296
00:19:06,370 --> 00:19:10,330
So the eigenvalues
are three and one --
297
00:19:10,330 --> 00:19:11,940
for this for sure.
298
00:19:11,940 --> 00:19:15,520
Now did we -- do you see why the
eigenvalues are three and one
299
00:19:15,520 --> 00:19:17,930
for that one?
300
00:19:17,930 --> 00:19:21,870
If I tell you the eigenvalues
are three and one, you prick --
301
00:19:21,870 --> 00:19:26,190
quickly process the trace,
which is -- and four --
302
00:19:26,190 --> 00:19:29,510
agrees with four and you
process the determinant,
303
00:19:29,510 --> 00:19:31,990
three times one --
304
00:19:31,990 --> 00:19:35,930
the determinant is three
and you say yes, it's right.
305
00:19:35,930 --> 00:19:39,540
Now I'm hoping that the
eigenvalues of this thing
306
00:19:39,540 --> 00:19:41,770
are three and one.
307
00:19:41,770 --> 00:19:45,450
May I process the trace and
the determinant for that one?
308
00:19:45,450 --> 00:19:47,830
What's the trace here?
309
00:19:47,830 --> 00:19:52,510
The trace of this matrix
is four minus two and six,
310
00:19:52,510 --> 00:19:54,320
and that's what it should be.
311
00:19:54,320 --> 00:19:59,210
What's the determinant minus
twelve plus fifteen is three.
312
00:19:59,210 --> 00:20:00,390
The determinant is three.
313
00:20:00,390 --> 00:20:04,190
The eigenvalues of that
matrix are also three and one.
314
00:20:04,190 --> 00:20:07,460
And you see I created
this matrix just like --
315
00:20:07,460 --> 00:20:11,600
I just took any M, like,
one that popped into my head
316
00:20:11,600 --> 00:20:16,030
and computed M inverse
A M, got that matrix,
317
00:20:16,030 --> 00:20:22,980
it didn't look anything
special but it's --
318
00:20:22,980 --> 00:20:26,290
like A itself, it has those
eigenvalues three and one.
319
00:20:26,290 --> 00:20:31,370
So that's the main fact
and let me write it down.
320
00:20:31,370 --> 00:20:45,290
Similar matrixes have
the same eigenvalues.
321
00:20:45,290 --> 00:20:51,590
So I'll just put that
as an important point.
322
00:20:51,590 --> 00:20:53,890
And think why.
323
00:20:56,690 --> 00:20:57,430
Why is that?
324
00:20:57,430 --> 00:21:01,120
So that's what that
family of matrixes is.
325
00:21:01,120 --> 00:21:03,940
The matrixes that
are similar to this A
326
00:21:03,940 --> 00:21:09,610
here are all the matrixes with
eigenvalues three and one.
327
00:21:09,610 --> 00:21:12,450
Every matrix with
eigenvalues three and one,
328
00:21:12,450 --> 00:21:16,870
there's some M that
connects this guy
329
00:21:16,870 --> 00:21:19,810
to the one you think of.
330
00:21:19,810 --> 00:21:22,840
And then of course, the most
special guy in the whole family
331
00:21:22,840 --> 00:21:26,690
is the diagonal one with
eigenvalues three and one
332
00:21:26,690 --> 00:21:28,630
sitting there on the diagonal.
333
00:21:28,630 --> 00:21:30,590
But also, I could find --
334
00:21:30,590 --> 00:21:34,130
I mean, tell me just a couple
more members of the family.
335
00:21:34,130 --> 00:21:38,070
Another -- tell me another
matrix that has eigenvalues
336
00:21:38,070 --> 00:21:38,680
three and one.
337
00:21:41,450 --> 00:21:44,740
Well, let's see, I -- oh,
I'll just make it triangular.
338
00:21:47,610 --> 00:21:49,180
That's in the family.
339
00:21:49,180 --> 00:21:53,840
There is some M that --
that connects to this one.
340
00:21:53,840 --> 00:21:58,310
And -- and also this.
341
00:21:58,310 --> 00:22:02,630
There's some matrix M -- so that
M inverse A M comes out to be
342
00:22:02,630 --> 00:22:03,180
that.
343
00:22:03,180 --> 00:22:06,290
There's a whole family here.
344
00:22:06,290 --> 00:22:11,710
And they all share
the same eigenvalues.
345
00:22:11,710 --> 00:22:14,320
So why is that?
346
00:22:14,320 --> 00:22:14,930
Okay.
347
00:22:14,930 --> 00:22:21,880
I'm going to start -- the only
possibility is to start with Ax
348
00:22:21,880 --> 00:22:24,180
equal lambda x.
349
00:22:24,180 --> 00:22:27,790
Okay, so suppose A has
the eigenvalue lambda.
350
00:22:30,970 --> 00:22:33,880
Now I want to get B into
the picture here somehow.
351
00:22:33,880 --> 00:22:37,930
You remember B is M inverse A M.
352
00:22:37,930 --> 00:22:39,910
Let's just remember
that over here.
353
00:22:39,910 --> 00:22:44,390
B is M inverse A M.
354
00:22:44,390 --> 00:22:48,030
And I want to see
its eigenvalues.
355
00:22:48,030 --> 00:22:51,520
How I going to get M inverse
A M into this equation?
356
00:22:51,520 --> 00:22:54,830
Let me just sort of do it.
357
00:22:54,830 --> 00:22:59,970
I'll put an M times an M
inverse in there, right?
358
00:22:59,970 --> 00:23:01,870
That was --
359
00:23:01,870 --> 00:23:04,100
I haven't changed
the left-hand side,
360
00:23:04,100 --> 00:23:06,420
so I better not change
the right-hand side.
361
00:23:09,570 --> 00:23:12,850
So everybody's okay so far,
I just put in there -- see,
362
00:23:12,850 --> 00:23:16,240
I want to get a -- so now
I'll multiply on the left by M
363
00:23:16,240 --> 00:23:17,870
inverse --
364
00:23:17,870 --> 00:23:20,200
I have to do the
same to this side
365
00:23:20,200 --> 00:23:22,880
and that number
lambda's just a number,
366
00:23:22,880 --> 00:23:25,380
so it factors out in the front.
367
00:23:25,380 --> 00:23:32,370
So what I have here
is this was safe.
368
00:23:32,370 --> 00:23:34,760
I did the same
thing to both sides.
369
00:23:34,760 --> 00:23:37,090
And now I've got B.
370
00:23:37,090 --> 00:23:38,270
There's B.
371
00:23:38,270 --> 00:23:42,210
That's B times this
vector M inverse
372
00:23:42,210 --> 00:23:47,670
x is equal to lambda times
this vector M inverse x.
373
00:23:47,670 --> 00:23:49,920
So what have I learned?
374
00:23:49,920 --> 00:23:53,980
I've learned that
B times some vector
375
00:23:53,980 --> 00:23:55,600
is lambda times that vector.
376
00:23:55,600 --> 00:23:58,910
I've learned that lambda
is an eigenvalue of B also.
377
00:23:58,910 --> 00:24:01,370
So this is -- if
-- so this is --
378
00:24:01,370 --> 00:24:05,600
if lambda's an eigenvalue of
A, then I can write it this way
379
00:24:05,600 --> 00:24:10,110
and I discover that
lambda's an eigenvalue of B.
380
00:24:10,110 --> 00:24:12,240
That's the end of the proof.
381
00:24:12,240 --> 00:24:16,670
The eigenvector
didn't stay the same.
382
00:24:16,670 --> 00:24:19,410
Of course I don't expect the
eigenvectors to stay the same.
383
00:24:19,410 --> 00:24:22,240
If all the eigenvalues are the
same and all the eigenvectors
384
00:24:22,240 --> 00:24:25,120
are the same, then probably
the matrix is the same.
385
00:24:27,960 --> 00:24:31,260
Here the eigenvector changes,
so the eigenvector --
386
00:24:31,260 --> 00:24:38,370
so the point is then
the eigenvector of B --
387
00:24:38,370 --> 00:24:44,880
of B is M inverse times
the eigenvector of A.
388
00:24:44,880 --> 00:24:45,380
Okay.
389
00:24:51,989 --> 00:24:53,280
That's all that this says here.
390
00:24:53,280 --> 00:24:58,940
The eigenvector of A was
X, and so the M inverse --
391
00:24:58,940 --> 00:25:01,930
similar matrixes, then
have the same eigenvalues
392
00:25:01,930 --> 00:25:05,070
and their eigenvectors
are just moved around.
393
00:25:05,070 --> 00:25:08,790
Of course, that's what we --
that's what happened way back
394
00:25:08,790 --> 00:25:09,580
--
395
00:25:09,580 --> 00:25:15,070
and the most important similar
matrixes are to diagonalize.
396
00:25:15,070 --> 00:25:18,140
So what was the point
when we diagonalized?
397
00:25:18,140 --> 00:25:20,830
The eigenvalues stayed
the same, of course.
398
00:25:20,830 --> 00:25:22,380
Three and one.
399
00:25:22,380 --> 00:25:25,150
What about the eigenvectors?
400
00:25:25,150 --> 00:25:29,190
The eigenvectors were whatever
they were for the matrix A,
401
00:25:29,190 --> 00:25:31,340
but then what were
the eigenvectors
402
00:25:31,340 --> 00:25:34,630
for the diagonal matrix?
403
00:25:34,630 --> 00:25:37,370
They're just -- what are the
eigenvectors of a diagonal
404
00:25:37,370 --> 00:25:37,900
matrix?
405
00:25:37,900 --> 00:25:41,130
They're just one
zero and zero one.
406
00:25:41,130 --> 00:25:44,290
So this step made the
eigenvectors nice,
407
00:25:44,290 --> 00:25:49,750
didn't change the
eigenvalues, and every time we
408
00:25:49,750 --> 00:25:51,580
don't change the eigenvalues.
409
00:25:51,580 --> 00:25:53,021
Same eigenvalues.
410
00:25:53,021 --> 00:25:53,520
Okay.
411
00:25:56,150 --> 00:26:01,100
Now -- so I've got all
these matrixes in --
412
00:26:01,100 --> 00:26:07,380
I've got this family of matrixes
with eigenvalues three and one.
413
00:26:07,380 --> 00:26:08,610
Fine.
414
00:26:08,610 --> 00:26:10,270
That's a nice family.
415
00:26:10,270 --> 00:26:15,520
It's nice because those two
eigenvalues are different.
416
00:26:15,520 --> 00:26:17,280
I now have to --
417
00:26:17,280 --> 00:26:19,460
to get into that --
418
00:26:19,460 --> 00:26:25,320
the -- into the less happy
possibility that the two
419
00:26:25,320 --> 00:26:26,960
eigenvalues could be the
420
00:26:26,960 --> 00:26:29,680
same.
421
00:26:29,680 --> 00:26:32,940
And then it's a little
trickier, because you remember
422
00:26:32,940 --> 00:26:35,360
when two eigenvalues
are the same,
423
00:26:35,360 --> 00:26:38,400
what's the bad possibility?
424
00:26:38,400 --> 00:26:42,030
That there might
not be enough --
425
00:26:42,030 --> 00:26:44,730
a full set of eigenvectors
and we might not be able
426
00:26:44,730 --> 00:26:46,080
to diagonalize.
427
00:26:46,080 --> 00:26:50,550
So I need to discuss
the bad case.
428
00:26:50,550 --> 00:26:53,070
So the bad -- can
I just say bad?
429
00:26:57,500 --> 00:27:03,707
If lambda one equals
lambda two, then the matrix
430
00:27:03,707 --> 00:27:04,873
might not be diagonalizable.
431
00:27:07,410 --> 00:27:11,010
Suppose lambda one equals
lambda two equals four,
432
00:27:11,010 --> 00:27:11,510
say.
433
00:27:14,460 --> 00:27:20,990
Now if I look at the family of
matrixes with eigenvalues four
434
00:27:20,990 --> 00:27:25,840
and four, well, one
possibility occurs to me.
435
00:27:25,840 --> 00:27:38,120
One family with eigenvalues four
and four has this matrix in it,
436
00:27:38,120 --> 00:27:41,420
four times the identity.
437
00:27:41,420 --> 00:27:46,400
Then another -- but now I want
to ask also about the matrix
438
00:27:46,400 --> 00:27:48,930
four four one zero.
439
00:27:51,800 --> 00:27:55,290
And my point -- here's
the whole point of this --
440
00:27:55,290 --> 00:28:00,107
of this bad stuff, is that this
guy is not in the same family
441
00:28:00,107 --> 00:28:00,690
with that one.
442
00:28:00,690 --> 00:28:05,310
The family of a -- of matrixes
that have eigenvalues four
443
00:28:05,310 --> 00:28:08,760
and four is two families.
444
00:28:08,760 --> 00:28:16,081
There's this total loner
here who's in a family off --
445
00:28:16,081 --> 00:28:16,580
right?
446
00:28:16,580 --> 00:28:18,890
Just by himself.
447
00:28:18,890 --> 00:28:23,070
And all the others
are in with this guy.
448
00:28:23,070 --> 00:28:35,090
So the big family
includes this one.
449
00:28:35,090 --> 00:28:39,720
And it includes a whole lot
of other matrixes, all --
450
00:28:39,720 --> 00:28:44,310
in fact, in this two by two
case, it -- you see where --
451
00:28:44,310 --> 00:28:48,290
what do I mean -- so what I
using, this word family --
452
00:28:48,290 --> 00:28:51,980
in a family, I mean
they're similar.
453
00:28:51,980 --> 00:28:56,570
So my point is that the only
matrix that's similar to this
454
00:28:56,570 --> 00:28:58,840
is itself.
455
00:28:58,840 --> 00:29:02,240
The only matrix that's similar
to four times the identity
456
00:29:02,240 --> 00:29:03,650
is four times the identity.
457
00:29:03,650 --> 00:29:05,320
It's off by itself.
458
00:29:05,320 --> 00:29:07,360
Why is that?
459
00:29:07,360 --> 00:29:11,750
The -- if this is my matrix,
four times the identity,
460
00:29:11,750 --> 00:29:17,150
and I take it, I multiply on
the right by any matrix M,
461
00:29:17,150 --> 00:29:22,430
I multiply on the left by
M inverse, what do I get?
462
00:29:22,430 --> 00:29:28,720
This is any M, but
what's the result?
463
00:29:28,720 --> 00:29:30,930
Well, factoring
out a four, that's
464
00:29:30,930 --> 00:29:34,710
just the identity
matrix in there.
465
00:29:34,710 --> 00:29:37,250
So then the M inverse
cancels the M,
466
00:29:37,250 --> 00:29:39,420
so I've just got this
matrix back again.
467
00:29:42,910 --> 00:29:45,310
So whatever the M
is, I'm not getting
468
00:29:45,310 --> 00:29:47,610
any more members of the family.
469
00:29:47,610 --> 00:29:57,680
So this is one small family,
because it only has one person.
470
00:29:57,680 --> 00:29:59,610
One matrix, excuse me.
471
00:29:59,610 --> 00:30:02,200
I think of these matrixes
as people by this point,
472
00:30:02,200 --> 00:30:04,110
in eighteen oh six.
473
00:30:04,110 --> 00:30:08,990
Okay, the other family
includes all the rest --
474
00:30:08,990 --> 00:30:13,930
all other matrixes that have
eigenvalues four and four.
475
00:30:13,930 --> 00:30:20,290
This is somehow the
best one in that family.
476
00:30:20,290 --> 00:30:21,870
See, I can't make it diagonal.
477
00:30:21,870 --> 00:30:24,580
If I -- if it's
diagonal, it's this one.
478
00:30:24,580 --> 00:30:27,020
It's in its own, by itself.
479
00:30:27,020 --> 00:30:29,250
So I have to think, okay,
what's the nearest I
480
00:30:29,250 --> 00:30:32,480
can get to diagonal?
481
00:30:32,480 --> 00:30:36,380
But it will not
be diagonalizable.
482
00:30:36,380 --> 00:30:39,670
That -- do you know that that
matrix is not diagonalizable?
483
00:30:39,670 --> 00:30:42,050
Of course, because if
it was diagonalizable,
484
00:30:42,050 --> 00:30:46,150
it would be similar to
that, which it isn't.
485
00:30:46,150 --> 00:30:48,540
The eigenvalues of
this are four and four,
486
00:30:48,540 --> 00:30:52,960
but what's the catch
with that matrix?
487
00:30:52,960 --> 00:30:55,430
It's only got one eigenvector.
488
00:30:55,430 --> 00:30:57,560
That's a
non-diagonalizable matrix.
489
00:30:57,560 --> 00:30:59,340
Only one eigenvector.
490
00:30:59,340 --> 00:31:06,700
And somehow, if I made that
one into a ten or to a million,
491
00:31:06,700 --> 00:31:10,540
I could find an M, it's in
the family, it's similar.
492
00:31:10,540 --> 00:31:14,570
But the best -- so the best
guy in this family is this one.
493
00:31:14,570 --> 00:31:19,810
And this is called the Jordan --
494
00:31:19,810 --> 00:31:24,800
so this guy Jordan picked
out -- so he, like, studied,
495
00:31:24,800 --> 00:31:29,250
these families of
matrixes, and each family,
496
00:31:29,250 --> 00:31:34,740
he picked out the nicest,
most diagonal one.
497
00:31:34,740 --> 00:31:37,960
But not completely diagonal,
because there's nobody --
498
00:31:37,960 --> 00:31:40,430
there isn't a diagonal
matrix in this family,
499
00:31:40,430 --> 00:31:44,780
so there's a one up
there in the Jordan form.
500
00:31:44,780 --> 00:31:46,010
Okay.
501
00:31:46,010 --> 00:31:50,699
I think we've got to see some
more matrixes in that family.
502
00:31:50,699 --> 00:31:52,990
So, all right, let me --
let's just think of some other
503
00:31:52,990 --> 00:31:58,570
matrixes whose eigenvalues are
four and four but they're not
504
00:31:58,570 --> 00:32:01,220
four times the identity.
505
00:32:01,220 --> 00:32:03,130
So -- and I believe that --
506
00:32:03,130 --> 00:32:06,660
that this -- that all the
examples we pick up will be
507
00:32:06,660 --> 00:32:13,320
similar to each other
and -- do you see why --
508
00:32:13,320 --> 00:32:17,340
in this topic of
similar matrixes,
509
00:32:17,340 --> 00:32:21,820
the climax is the Jordan form.
510
00:32:21,820 --> 00:32:23,910
So it says that every matrix --
511
00:32:23,910 --> 00:32:29,310
I'll write down what the Jordan
form -- what Jordan discovered.
512
00:32:29,310 --> 00:32:35,030
He found the best looking
matrix in each family.
513
00:32:35,030 --> 00:32:41,080
And that's -- then we've got --
then we've covered all matrixes
514
00:32:41,080 --> 00:32:44,540
including the
non-diagonalizable one.
515
00:32:44,540 --> 00:32:47,060
That -- that's the
point, that in some way,
516
00:32:47,060 --> 00:32:50,430
Jordan completed the
diagonalization by coming
517
00:32:50,430 --> 00:32:54,890
as near as he could,
which is his Jordan form.
518
00:32:54,890 --> 00:32:57,340
And therefore, if you want
to cover all matrixes,
519
00:32:57,340 --> 00:32:59,720
you've got to get
him in the picture.
520
00:32:59,720 --> 00:33:03,030
It used to be -- when
I took eighteen oh six,
521
00:33:03,030 --> 00:33:07,530
that was the climax of the
course, this Jordan form stuff.
522
00:33:07,530 --> 00:33:11,490
I think it's not the climax
of linear algebra anymore,
523
00:33:11,490 --> 00:33:15,930
because --
524
00:33:15,930 --> 00:33:20,590
it's not easy to
find this Jordan form
525
00:33:20,590 --> 00:33:25,340
for a general matrix, because
it depends on these eigenvalues
526
00:33:25,340 --> 00:33:27,900
being exactly the same.
527
00:33:27,900 --> 00:33:30,620
You'd have to know exactly
the eigenvalues and it --
528
00:33:30,620 --> 00:33:35,270
and you'd have to know exactly
the rank and the slightest
529
00:33:35,270 --> 00:33:39,180
change in numbers will
change those eigenvalues,
530
00:33:39,180 --> 00:33:43,320
change the rank and therefore
the whole thing is numerically
531
00:33:43,320 --> 00:33:46,970
not an -- a good thing.
532
00:33:46,970 --> 00:33:51,180
But for algebra,
it's the right thing
533
00:33:51,180 --> 00:33:52,660
to understand this family.
534
00:33:52,660 --> 00:33:56,600
So just tell me another matrix
-- a few more matrixes --
535
00:33:56,600 --> 00:34:03,380
so more members of the family.
536
00:34:06,510 --> 00:34:11,380
Let me put down again
what the best one is.
537
00:34:11,380 --> 00:34:12,400
Okay.
538
00:34:12,400 --> 00:34:13,409
All right.
539
00:34:13,409 --> 00:34:14,690
Some more matrixes.
540
00:34:14,690 --> 00:34:17,659
Let's see, what I looking for?
541
00:34:17,659 --> 00:34:22,750
I'm looking for matrixes
whose trace is what?
542
00:34:22,750 --> 00:34:25,190
So if I'm looking for more
matrixes in the family,
543
00:34:25,190 --> 00:34:28,290
they'll all have the same
eigenvalues, four and four.
544
00:34:28,290 --> 00:34:30,449
So their trace will be eight.
545
00:34:30,449 --> 00:34:34,120
So why don't I just take,
like, five and three --
546
00:34:34,120 --> 00:34:40,540
I've got the trace right, now
the determinant should be what?
547
00:34:40,540 --> 00:34:41,469
Sixteen.
548
00:34:41,469 --> 00:34:45,370
So I just fix this up -- shall I
put maybe a one and a minus one
549
00:34:45,370 --> 00:34:46,771
there?
550
00:34:46,771 --> 00:34:47,270
Okay.
551
00:34:47,270 --> 00:34:52,139
There's a matrix with
eigenvalues four and four,
552
00:34:52,139 --> 00:34:56,719
because the trace is eight and
the determinant is sixteen.
553
00:34:56,719 --> 00:35:00,670
And I don't think
it's diagonalizable.
554
00:35:00,670 --> 00:35:03,800
Do you know why it's
not diagonalizable?
555
00:35:03,800 --> 00:35:06,470
Because if it was
diagonalizable,
556
00:35:06,470 --> 00:35:09,505
the diagonal form
would have to be this.
557
00:35:12,460 --> 00:35:14,980
But I can't get to that
form, because whatever
558
00:35:14,980 --> 00:35:17,880
I do with any M inverse and
M I stay with that form.
559
00:35:17,880 --> 00:35:20,320
I could never get
-- connect those.
560
00:35:20,320 --> 00:35:22,470
So I can put down more
members -- here --
561
00:35:22,470 --> 00:35:23,810
here's another easy one.
562
00:35:23,810 --> 00:35:26,960
I could put the four and
the four and a seventeen
563
00:35:26,960 --> 00:35:27,580
down there.
564
00:35:30,270 --> 00:35:32,080
All these matrixes are similar.
565
00:35:32,080 --> 00:35:35,350
If I'm -- I could find an M
that would show that that one is
566
00:35:35,350 --> 00:35:37,350
similar to that one.
567
00:35:37,350 --> 00:35:40,420
And in -- you can see the
general picture is I can take
568
00:35:40,420 --> 00:35:45,710
any a and any 8-a here and
any -- oh, I don't know,
569
00:35:45,710 --> 00:35:48,200
whatever you put it'd be
-- anyway, you can see.
570
00:35:48,200 --> 00:35:54,890
I can fill this in, fill this in
to make the trace equal eight,
571
00:35:54,890 --> 00:36:01,160
the determinant equal 16, I
get all that family of matrixes
572
00:36:01,160 --> 00:36:02,635
and they're all similar.
573
00:36:05,170 --> 00:36:08,250
So we see what eigenvalues do.
574
00:36:08,250 --> 00:36:13,020
They're all similar and they
all have only one eigenvector.
575
00:36:13,020 --> 00:36:16,610
So I -- if I'm -- if
you were going to --
576
00:36:16,610 --> 00:36:20,780
allow me to add to this picture,
they have the same lambdas
577
00:36:20,780 --> 00:36:25,799
and they also have the
same number of independent
578
00:36:25,799 --> 00:36:26,340
eigenvectors.
579
00:36:29,880 --> 00:36:33,910
Because if I get an eigenvector
for x I get one for -- for A,
580
00:36:33,910 --> 00:36:36,260
I get one for B also.
581
00:36:36,260 --> 00:36:43,170
So -- and same number
of eigenvectors.
582
00:36:43,170 --> 00:36:45,740
But even more than that --
583
00:36:45,740 --> 00:36:47,048
even more than that --
584
00:36:47,048 --> 00:36:49,173
I mean, it's not enough
just to count eigenvectors.
585
00:36:51,970 --> 00:36:54,450
Yes, let me give you an
example why it's not even
586
00:36:54,450 --> 00:36:58,460
enough to count eigenvectors.
587
00:36:58,460 --> 00:37:00,170
So another example.
588
00:37:00,170 --> 00:37:04,210
So here are some matrixes --
589
00:37:04,210 --> 00:37:07,600
oh, let me make
them four by four --
590
00:37:07,600 --> 00:37:09,260
okay, here -- here's a matrix.
591
00:37:09,260 --> 00:37:11,570
I mean, like if you
want nightmares,
592
00:37:11,570 --> 00:37:14,480
think about matrixes like these.
593
00:37:14,480 --> 00:37:21,670
Uh, so a one off the diagonal
-- say a one there, how many --
594
00:37:21,670 --> 00:37:24,600
what are the eigenvalues
of that matrix?
595
00:37:24,600 --> 00:37:29,040
Oh, I mean --
596
00:37:29,040 --> 00:37:30,480
okay.
597
00:37:30,480 --> 00:37:32,550
What are the eigenvalues
of that matrix?
598
00:37:35,740 --> 00:37:37,910
Please.
599
00:37:37,910 --> 00:37:40,160
Four 0s, right?
600
00:37:40,160 --> 00:37:44,930
So we're really getting
bad matrixes now.
601
00:37:44,930 --> 00:37:47,080
So I mean, this is, like --
602
00:37:47,080 --> 00:37:53,190
Jordan was a good guy, but he
had to think about matrixes
603
00:37:53,190 --> 00:37:58,590
that all -- that had, like -- an
eigenvalue repeated four times.
604
00:37:58,590 --> 00:38:01,140
How many eigenvectors
does that matrix have?
605
00:38:04,750 --> 00:38:07,980
Well, I'm --
eigenvectors will be --
606
00:38:07,980 --> 00:38:11,710
since the eigenvalue is
zero, eigenvectors will be
607
00:38:11,710 --> 00:38:13,350
in the null space, right?
608
00:38:13,350 --> 00:38:17,870
I'm -- eigenvectors have
got to be A x equal zero x.
609
00:38:17,870 --> 00:38:21,340
So what's the dimension
of the null space?
610
00:38:21,340 --> 00:38:22,110
Two.
611
00:38:22,110 --> 00:38:23,920
Somebody said two.
612
00:38:23,920 --> 00:38:24,680
And that's right.
613
00:38:24,680 --> 00:38:26,690
How -- why?
614
00:38:26,690 --> 00:38:29,390
Because you ask what's
the rank of that matrix,
615
00:38:29,390 --> 00:38:32,410
the rank is obviously two.
616
00:38:32,410 --> 00:38:35,050
The number of
independent rows is two,
617
00:38:35,050 --> 00:38:36,880
the number of independent
columns is two,
618
00:38:36,880 --> 00:38:41,430
the rank is two so the null --
the dimension of the null space
619
00:38:41,430 --> 00:38:45,950
is four minus two, so
it's got two eigenvectors.
620
00:38:45,950 --> 00:38:47,500
Two eigenvectors.
621
00:38:47,500 --> 00:38:49,441
Two independent eigenvectors.
622
00:38:49,441 --> 00:38:49,940
All right.
623
00:38:49,940 --> 00:38:55,260
The dimension of the
null space is two.
624
00:38:59,910 --> 00:39:03,905
Now, suppose I change
this zero to a seven.
625
00:39:09,050 --> 00:39:11,840
The eigenvalues are all still
zero, how -- what about --
626
00:39:11,840 --> 00:39:12,756
how many eigenvectors?
627
00:39:15,449 --> 00:39:17,990
What's the dimension of the --
what's the rank of this matrix
628
00:39:17,990 --> 00:39:19,160
now?
629
00:39:19,160 --> 00:39:20,890
Still two, right?
630
00:39:20,890 --> 00:39:22,820
So it's okay.
631
00:39:22,820 --> 00:39:26,370
And actually, this would
be similar to the one that
632
00:39:26,370 --> 00:39:27,800
had a zero in there.
633
00:39:27,800 --> 00:39:31,740
But it's not as beautiful,
Jordan picked this one.
634
00:39:31,740 --> 00:39:34,850
He picked -- he put ones --
635
00:39:34,850 --> 00:39:39,000
we have a one on the -- above
the diagonal for every missing
636
00:39:39,000 --> 00:39:42,330
eigenvector, and here we're
missing two because we've got
637
00:39:42,330 --> 00:39:45,550
two, so we've got two
eigenvectors and two are
638
00:39:45,550 --> 00:39:53,860
missing, because it's
a four by four matrix.
639
00:39:53,860 --> 00:39:58,910
Okay, now -- but I was going to
give you this second example.
640
00:40:01,690 --> 00:40:05,145
0 1 0 0, let me
just move the one.
641
00:40:09,030 --> 00:40:11,500
Oop, not there.
642
00:40:11,500 --> 00:40:15,710
Off the diagonal and
zero zero zero zero zero.
643
00:40:15,710 --> 00:40:16,210
Okay.
644
00:40:19,420 --> 00:40:22,570
So now tell me
about this matrix.
645
00:40:22,570 --> 00:40:27,460
Its eigenvalues are
four zeroes again.
646
00:40:27,460 --> 00:40:31,910
Its rank is two again.
647
00:40:31,910 --> 00:40:36,670
So it has two eigenvectors
and two missing.
648
00:40:36,670 --> 00:40:40,890
But the darn thing is
not similar to that one.
649
00:40:40,890 --> 00:40:44,430
A -- a count of eigenvectors
looks like these could be
650
00:40:44,430 --> 00:40:47,130
similar, but they're not.
651
00:40:47,130 --> 00:40:52,720
Jordan -- see, this is like --
a little three by three block
652
00:40:52,720 --> 00:40:55,880
and a little one by one block.
653
00:40:55,880 --> 00:40:58,770
And this one is like a
two by two block and a two
654
00:40:58,770 --> 00:41:03,080
by two block, and those blocks
are called Jordan blocks.
655
00:41:03,080 --> 00:41:06,770
So let me say what
is a Jordan block?
656
00:41:11,150 --> 00:41:17,400
J block number I has --
657
00:41:17,400 --> 00:41:22,660
so a Jordan block has a repeated
eigenvalue, lambda I, lambda I
658
00:41:22,660 --> 00:41:24,530
on the diagonal.
659
00:41:24,530 --> 00:41:26,905
Zeroes below and ones above.
660
00:41:30,060 --> 00:41:34,160
So there's a block
with this guy repeated,
661
00:41:34,160 --> 00:41:37,070
but it only has one eigenvector.
662
00:41:37,070 --> 00:41:40,900
So a Jordan block has
one eigenvector only.
663
00:41:43,970 --> 00:41:47,720
This one has one eigenvector,
this block has one eigenvector
664
00:41:47,720 --> 00:41:49,580
and we get two.
665
00:41:49,580 --> 00:41:52,170
This block has one
eigenvector and that block has
666
00:41:52,170 --> 00:41:54,980
one eigenvector and we get two.
667
00:41:54,980 --> 00:42:02,020
So -- but the blocks
are different sizes.
668
00:42:02,020 --> 00:42:05,540
And that -- it turns
out Jordan worked out --
669
00:42:05,540 --> 00:42:14,190
then this is not similar,
not similar to this one.
670
00:42:18,020 --> 00:42:22,140
So the -- so I'm, like,
giving you the whole story --
671
00:42:22,140 --> 00:42:25,780
well, not the whole story, but
the main themes of the story --
672
00:42:25,780 --> 00:42:29,700
is here's Jordan's theorem.
673
00:42:29,700 --> 00:42:49,060
Every square matrix A is
similar to A Jordan matrix J.
674
00:42:49,060 --> 00:42:51,890
And what's a Jordan matrix J?
675
00:42:51,890 --> 00:42:56,670
It's a matrix with
these blocks, block --
676
00:42:56,670 --> 00:43:03,850
Jordan block number one, Jordan
block number two and so on.
677
00:43:03,850 --> 00:43:07,950
And let's say Jordan
block number d.
678
00:43:11,360 --> 00:43:14,760
And those Jordan
blocks look like that,
679
00:43:14,760 --> 00:43:17,760
so the eigenvalues are
sitting on the diagonal,
680
00:43:17,760 --> 00:43:22,750
but we've got some of these
ones above the diagonal.
681
00:43:22,750 --> 00:43:25,010
We've got the number of --
682
00:43:25,010 --> 00:43:27,530
so the number of blocks --
683
00:43:27,530 --> 00:43:37,510
the number of blocks is
the number of eigenvectors,
684
00:43:37,510 --> 00:43:42,510
because we get one
eigenvector per block.
685
00:43:42,510 --> 00:43:47,250
So what I'm -- so if I
summarize Jordan's idea --
686
00:43:47,250 --> 00:43:49,570
start with any A.
687
00:43:49,570 --> 00:43:54,080
If its eigenvalues are
distinct, then what's it similar
688
00:43:54,080 --> 00:43:54,700
to?
689
00:43:54,700 --> 00:43:56,430
This is the good case.
690
00:43:56,430 --> 00:44:00,710
if I start with a matrix A and
it has different eigenvalues --
691
00:44:00,710 --> 00:44:03,450
it's n eigenvalues, none
of them are repeated,
692
00:44:03,450 --> 00:44:09,350
then that's a diagonal --
diagonalizable matrix --
693
00:44:09,350 --> 00:44:13,990
the Jordan blocks is -- has --
the Jordan matrix is diagonal.
694
00:44:13,990 --> 00:44:15,900
It's lambda.
695
00:44:15,900 --> 00:44:17,840
So the good case --
696
00:44:17,840 --> 00:44:22,590
the good case, J is lambda.
697
00:44:27,020 --> 00:44:29,400
All -- there are --
698
00:44:29,400 --> 00:44:29,900
d=n.
699
00:44:29,900 --> 00:44:33,830
There are n eigenvectors, n
blocks, diagonal, everything
700
00:44:33,830 --> 00:44:35,640
great.
701
00:44:35,640 --> 00:44:40,990
But Jordan covered
all cases by including
702
00:44:40,990 --> 00:44:44,460
these cases of repeated
eigenvalues and missing
703
00:44:44,460 --> 00:44:47,100
eigenvectors.
704
00:44:47,100 --> 00:44:47,810
Okay.
705
00:44:47,810 --> 00:44:49,640
That's a description of Jordan.
706
00:44:49,640 --> 00:44:51,030
That -- that's --
707
00:44:51,030 --> 00:44:54,300
I haven't told you how
to compute this thing,
708
00:44:54,300 --> 00:44:56,500
and it isn't easy.
709
00:44:56,500 --> 00:45:00,920
Whereas the good case is the --
the good case is what 18.06 is
710
00:45:00,920 --> 00:45:01,660
about.
711
00:45:01,660 --> 00:45:07,040
The -- this case is what
18.06 was about 20 years ago.
712
00:45:07,040 --> 00:45:12,220
So you can see you probably
won't have on the final exam
713
00:45:12,220 --> 00:45:18,540
the computation of a Jordan
matrix for some horrible thing
714
00:45:18,540 --> 00:45:21,720
with four repeated eigenvalues.
715
00:45:21,720 --> 00:45:28,730
I'm not that crazy
about the Jordan form.
716
00:45:28,730 --> 00:45:34,950
But I'm very positive about
positive definite matrixes
717
00:45:34,950 --> 00:45:38,880
and about the idea
that's coming Monday,
718
00:45:38,880 --> 00:45:40,964
the singular value
decomposition.
719
00:45:40,964 --> 00:45:43,130
So I'll see you on Monday,
and have a great weekend.
720
00:45:43,130 --> 00:45:44,680
Bye.