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OK.
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So this is the first lecture on
eigenvalues and eigenvectors,
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and that's a big subject
that will take up
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most of the rest of the course.
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It's, again, matrices are
square and we're looking now
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for some special
numbers, the eigenvalues,
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and some special vectors,
the eigenvectors.
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And so this lecture is mostly
about what are these numbers,
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and then the other lectures
are about how do we use them,
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why do we want them.
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OK, so what's an eigenvector?
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Maybe I'll start
with eigenvector.
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What's an eigenvector?
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So I have a matrix A.
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OK.
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What does a matrix do?
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It acts on vectors.
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It multiplies vectors x.
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So the way that matrix acts
is in goes a vector x and out
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comes a vector Ax.
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It's like a function.
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With a function in
calculus, in goes
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a number x, out comes f(x).
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Here in linear algebra
we're up in more dimensions.
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In goes a vector x,
out comes a vector Ax.
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And the vectors I'm
specially interested in
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are the ones the come
out in the same direction
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that they went in.
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That won't be typical.
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Most vectors, Ax is in -- points
in some different direction.
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But there are certain vectors
where Ax comes out parallel
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to x.
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And those are the eigenvectors.
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So Ax parallel to x.
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Those are the eigenvectors.
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And what do I mean by parallel?
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Oh, much easier to just
state it in an equation.
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Ax is some multiple -- and
everybody calls that multiple
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lambda -- of x.
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That's our big equation.
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We look for special vectors
-- and remember most vectors
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won't be eigenvectors --
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that -- for which Ax is in
the same direction as x,
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and by same direction I allow
it to be the very opposite
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direction, I allow lambda
to be negative or zero.
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Well, I guess we've met
the eigenvectors that
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have eigenvalue zero.
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Those are in the same
direction, but they're --
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in a kind of very special way.
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So this -- the eigenvector x.
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Lambda, whatever this
multiplying factor
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is, whether it's six or
minus six or zero or even
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some imaginary number,
that's the eigenvalue.
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So there's the eigenvalue,
there's the eigenvector.
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Let's just take a second
on eigenvalue zero.
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From the point of view of
eigenvalues, that's no special
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deal.
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That's, we have an eigenvector.
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If the eigenvalue
happened to be zero,
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that would mean that Ax was
zero x, in other words zero.
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So what would x, where would we
look for -- what are the x-s?
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What are the eigenvectors
with eigenvalue zero?
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They're the guys in the
null space, Ax equals zero.
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So if our matrix is singular,
let me write this down.
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If, if A is singular, then that
-- what does singular mean?
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It means that it takes
some vector x into zero.
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Some non-zero
vector, that's why --
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will be the
eigenvector into zero.
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Then lambda equals
zero is an eigenvalue.
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But we're interested
in all eigenvalues now,
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lambda equals zero is not,
like, so special anymore.
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OK.
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So the question is, how do we
find these x-s and lambdas?
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And notice -- we don't have an
equation Ax equal B anymore.
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I can't use elimination.
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I've got two unknowns,
and in fact they're
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multiplied together.
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Lambda and x are
both unknowns here.
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So, we need to, we need a
good idea of how to find them.
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But before I, before
I do that, and that's
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where determinant will
come in, can I just
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give you some matrices?
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Like here you go.
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Take the matrix, a
projection matrix.
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OK.
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So suppose we have a plane
and our matrix P is --
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what I've called A, now
I'm going to call it P
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for the moment, because it's --
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I'm thinking OK, let's
look at a then this,
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this other new matrix, I just
have an Ax, projection matrix.
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What are the eigenvalues
of a projection matrix?
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So that's my question.
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What are the x-s, the
eigenvectors, and the lambdas,
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the eigenvalues, thing,4 but the
roots of that quadratic for --
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and now let me say
a projection matrix.
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My, my point is that we --
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before we get into
determinants and, and formulas
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and all that stuff,
let's take some matrices
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where we know what they do.
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We know that if we take a
vector b, what this matrix does
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is it projects it down to Pb.
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So is b an eigenvector
in, in that picture?
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Is that vector b an eigenvector?
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No.
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Not so, so b is
not an eigenvector
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c- because Pb, its projection,
is in a different direction.
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So now tell me what vectors
are eigenvectors of P?
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What vectors do get projected
in the same direction that they
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start?
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So, so answer, tell me some x-s.
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Do you see what3 so it's
if Ax equals lambda x,
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In this picture, where could
I start with a vector b or x,
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do its projection, and end
up in the same direction?
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Well, that would happen if the
vector was right in that plane
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already.
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If the vector x was --
so let the vector x --
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so any vector, any x in the
plane will be an eigenvector.
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And what will happen
when I multiply by P,
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when I project a vector x --
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I called it b here, because
this is our familiar picture,
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but now I'm going to say that
b was no good for, for the,
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for our purposes.
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I'm interested in a vector x
that's actually in the plane,
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and I project it, and
what do I get back?
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x, of course.
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Doesn't move. can
be complex numbers.
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So any x in the plane
is unchanged by P,
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and what's that telling me?
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That's telling me that
x is an eigenvector,
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and it's also telling me what's
the eigenvalue, which is --
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just compare it with that.
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The eigenvalue, the
multiplier, is just one.
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Good.
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So we have actually a whole
plane of eigenvectors.
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Now I ask, are there
any other eigenvectors?
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And I expect the
answer to be yes,
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because I would
like to get three,
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if I'm in three
dimensions, I would
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like to hope for three
independent eigenvectors, two
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of them in the plane and
one not in the plane.
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OK.
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So this guy b that I drew
there was not any good.
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What's the right eigenvector
that's not in the plane?
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The, the good one is the one
that's perpendicular to the
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plane.
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There's an, another good x,
because what's the projection?
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So these are eigenvectors.
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Another guy here would
be another eigenvector.
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But now here is
another one. two.
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Any x that's perpendicular
to the plane,
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what's Px for that,
for that, vector?
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What's the projection of this
guy perpendicular to the plane?
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It is zero, of course.
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So -- there's the null space.
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Px and n- for those guys are
zero, or zero x if we like,
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and the eigenvalue is zero.
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So my answer to the question
is, what are the eigenvalues for
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In our example, the
one we worked out,
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a projection matrix?
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There they are.
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One and zero.
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OK.
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We know projection matrices.
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We can write them down as that
A, A transpose, A inverse,
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A transpose thing, but without
doing that from the picture
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we could see what
are the eigenvectors.
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OK.
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Are there other matrices?
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Let me take a second example.
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How about a permutation matrix?
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What about the matrix,
I'll call it A now.
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Zero one, one zero.
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A equals zero one one zero,
that had eigenvalue one and
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Can you tell me a vector x --
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see, we'll have a
system soon enough,
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so I, I would like
to just do these e-
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these couple of examples, just
to see the picture before we,
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before we let it
all, go into a system
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where that, matrix
isn't anything special.
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Because it is special.
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And what, so what vector
could I multiply by and end up
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in the same direction?
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Can you spot an
eigenvector for this guy?
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That's a matrix that
permutes x1 and x2, right?
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It switches the two
components of x.
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How could the vector
with its x2 x1, with --
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permuted turn out to
be a multiple of x1 x2,
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the vector we start with?
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Can you tell me an
eigenvector here for this guy?
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x equal -- what is -- actually,
can you tell me one vector that
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which is lambda x,
and I have a three x,
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And of course you -- everybody
knows that they're -- what,
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has eigenvalue one?
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So what, what vector
would have eigenvalue one,
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just above what we2
found here. so that if I,
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if I permute it it
doesn't change? right?
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There, that could
be one one, thanks.
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One one.
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00:12:14,460 --> 00:12:16,780
OK, take that vector one one.
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That will be an eigenvector,
because if I do Ax
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I get one one.
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So that's the eigenvalue is one.
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Great.
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That's one eigenvalue.
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00:12:30,600 --> 00:12:33,420
But I have here a
two by two matrix,
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and I figure there's going
to be a second eigenvalue.
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And eigenvector.
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00:12:42,740 --> 00:12:44,410
Now, what about that?
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00:12:44,410 --> 00:12:50,280
What's a vector, OK, maybe
we can just, like, guess it.
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00:12:50,280 --> 00:12:55,280
A vector that the
other -- actually,
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00:12:55,280 --> 00:12:58,940
this one that I'm thinking of
is going to be a vector that has
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00:12:58,940 --> 00:13:00,630
eigenvalue minus one.
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00:13:03,240 --> 00:13:06,670
That's going to be my other
eigenvalue for this matrix.
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00:13:06,670 --> 00:13:09,430
It's a -- notice the nice
positive or not negative
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00:13:09,430 --> 00:13:13,240
matrix, but an eigenvalue is
going to come out negative.
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00:13:13,240 --> 00:13:15,520
And can you guess, spot
the x that will work for
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Times x is supposed to
give me zero, right? that?
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00:13:18,910 --> 00:13:20,960
So I want a, a vector.
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00:13:20,960 --> 00:13:27,140
When I multiply by A, which
reverses the two components,
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I want the thing to come
out minus the original.
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00:13:31,120 --> 00:13:34,480
So what shall I send
in in that case?
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00:13:34,480 --> 00:13:37,650
If I send in negative one one.
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Then when I apply A, I get
I do that multiplication,
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and I get one negative
one, so it reversed sign.
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So Ax is -x.
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00:13:55,840 --> 00:13:57,710
Lambda is minus one.
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Ax -- so Ax was x there
and Ax is minus x here.
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00:14:04,130 --> 00:14:08,080
Can I just mention,
like, jump ahead, have,
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00:14:08,080 --> 00:14:10,550
give a perfectly
innocent-looking quadratic
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00:14:10,550 --> 00:14:14,990
and point out a special
little fact about eigenvalues.
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00:14:14,990 --> 00:14:18,940
n by n matrices will
have n eigenvalues.
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And I get this matrix4
zero zero zero one,
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00:14:23,390 --> 00:14:28,390
And it's not like -- suppose
n is three or four or more.
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It's not so easy to find them.
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00:14:32,120 --> 00:14:35,570
We'd have a third degree or a
fourth degree or an n-th degree
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00:14:35,570 --> 00:14:36,330
equation.
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00:14:36,330 --> 00:14:37,960
But here's one nice fact.
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00:14:37,960 --> 00:14:40,270
There, there's one
pleasant fact. we --
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00:14:40,270 --> 00:14:42,570
the eigenvalues came
out four and two.
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00:14:42,570 --> 00:14:45,540
That the sum of the
eigenvalues equals the sum
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00:14:45,540 --> 00:14:46,530
down the diagonal.
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00:14:46,530 --> 00:14:50,150
That's called the trace, and
I put that in the lecture
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00:14:50,150 --> 00:14:54,590
Now I add three I to that
matrix. content specifically.
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00:14:54,590 --> 00:15:03,390
So this is a neat fact, the fact
that sthe sum of the lambdas,
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00:15:03,390 --> 00:15:08,390
add up the lambdas,
equals the sum --
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00:15:08,390 --> 00:15:11,220
what would you like me to,
shall I write that down?
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00:15:11,220 --> 00:15:17,330
What I'm want to say in words is
the sum down the diagonal of A.
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00:15:17,330 --> 00:15:18,640
Shall I write a11+a22+...+ ann.
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That's add up the
diagonal entries.
250
00:15:27,470 --> 00:15:31,700
In this example, it's zero.
251
00:15:31,700 --> 00:15:36,560
In other words, once I found
this eigenvalue of one,
252
00:15:36,560 --> 00:15:39,110
I knew the other one
had to be minus one
253
00:15:39,110 --> 00:15:43,710
in this two by two case, because
in the two by two case, which
254
00:15:43,710 --> 00:15:51,970
is a good one to, to, play
with, the trace tells you
255
00:15:51,970 --> 00:15:54,804
right away what the
other eigenvalue is.
256
00:15:54,804 --> 00:15:56,970
So if I tell you one
eigenvalue, you can tell me the
257
00:15:56,970 --> 00:15:57,780
other one.
258
00:15:57,780 --> 00:16:02,270
We'll, we'll have that -- we'll,
minus one and eigenvectors one
259
00:16:02,270 --> 00:16:05,850
one and eigenvector minus
one we'll see that again.
260
00:16:05,850 --> 00:16:06,350
OK.
261
00:16:06,350 --> 00:16:07,980
Now can I --
262
00:16:07,980 --> 00:16:11,240
I could give more
examples, but maybe it's
263
00:16:11,240 --> 00:16:16,140
time to face the, the equation,
Ax equal lambda x, and figure
264
00:16:16,140 --> 00:16:19,810
how are we going to
find x and lambda.
265
00:16:19,810 --> 00:16:22,571
And that is lambda
one times lambda3
266
00:16:22,571 --> 00:16:23,070
OK.
267
00:16:23,070 --> 00:16:25,590
So this, so the
question now is how
268
00:16:25,590 --> 00:16:30,630
to find eigenvalues
and eigenvectors.
269
00:16:30,630 --> 00:16:35,430
How to solve, how to solve Ax
equal lambda x from the three
270
00:16:35,430 --> 00:16:43,670
x, so it's just I mean,
when we've got two unknowns
271
00:16:43,670 --> 00:16:47,500
both in the equation.
272
00:16:47,500 --> 00:16:48,460
OK.
273
00:16:48,460 --> 00:16:51,330
Here's the trick.
274
00:16:51,330 --> 00:16:53,240
Simple idea.
275
00:16:53,240 --> 00:16:58,980
Bring this onto the same side.
276
00:16:58,980 --> 00:16:59,950
Rewrite.
277
00:16:59,950 --> 00:17:05,589
Bring this over as A minus
lambda times the identity x
278
00:17:05,589 --> 00:17:09,770
One. equals zero.
279
00:17:09,770 --> 00:17:10,740
Right?
280
00:17:10,740 --> 00:17:14,150
I have Ax minus lambda
x, so I brought that over
281
00:17:14,150 --> 00:17:14,260
and I've got zero left on
the, on the right-hand side.
282
00:17:14,260 --> 00:17:16,740
What's the relation
between that problem and --
283
00:17:16,740 --> 00:17:19,170
let me write
284
00:17:19,170 --> 00:17:19,980
OK.
285
00:17:19,980 --> 00:17:23,200
I don't know lambda and I don't
know x, but I do know something
286
00:17:23,200 --> 00:17:23,700
here.
287
00:17:26,490 --> 00:17:28,460
What I know is if
I, if I'm going
288
00:17:28,460 --> 00:17:31,870
to be able to solve this thing,
for some x that's not the zero
289
00:17:31,870 --> 00:17:35,760
vector, that's not, that's
a useless eigenvector,
290
00:17:35,760 --> 00:17:37,410
doesn't count.
291
00:17:37,410 --> 00:17:44,860
What I know now is that
this matrix must be what?
292
00:17:44,860 --> 00:17:48,350
If I'm going to be --
if there is an x --
293
00:17:48,350 --> 00:17:51,520
I don't -- right now I
don't know what it is.
294
00:17:51,520 --> 00:17:54,490
I'm going to find
lambda first, actually.
295
00:17:54,490 --> 00:18:00,300
And -- but if there is an x,
it tells me that this matrix,
296
00:18:00,300 --> 00:18:05,160
this special combination,
which is like the matrix A with
297
00:18:05,160 --> 00:18:10,430
lambda -- shifted by
lambda, shifted by lambda I,
298
00:18:10,430 --> 00:18:14,550
that it has to be singular.
299
00:18:14,550 --> 00:18:17,030
This matrix must be
singular, otherwise
300
00:18:17,030 --> 00:18:21,510
the only x would be the
zero x, and zero matrix.OK.
301
00:18:21,510 --> 00:18:22,700
So this is singular.
302
00:18:22,700 --> 00:18:30,320
And what do I now know
about singular matrices?
303
00:18:30,320 --> 00:18:31,520
So, so take three away.
304
00:18:31,520 --> 00:18:35,140
Their determinant is zero.
305
00:18:35,140 --> 00:18:38,790
So I've -- so from the fact
that that has to be singular,
306
00:18:38,790 --> 00:18:44,750
I know that the determinant of
A minus lambda I has to be zero.
307
00:18:48,590 --> 00:18:52,430
And that, now I've
got x out of it.
308
00:18:52,430 --> 00:18:55,680
I've got an equation for
lambda, that the key equation --
309
00:18:55,680 --> 00:19:01,020
it's called the characteristic
equation or the eigenvalue
310
00:19:01,020 --> 00:19:01,520
equation.
311
00:19:04,180 --> 00:19:08,370
And that -- in other words,
I'm now in a position to find
312
00:19:08,370 --> 00:19:10,480
lambda first.
313
00:19:10,480 --> 00:19:15,940
So -- the idea will be
to find lambda first.
314
00:19:18,890 --> 00:19:21,660
And actually, I won't
find one lambda,
315
00:19:21,660 --> 00:19:24,120
I'll find N different lambdas.
316
00:19:24,120 --> 00:19:28,440
Well, n lambdas, maybe
not n different ones.
317
00:19:28,440 --> 00:19:31,430
A lambda could be repeated.
318
00:19:31,430 --> 00:19:38,040
A repeated lambda is the
source of all trouble in 18.06.
319
00:19:38,040 --> 00:19:43,620
So, let's hope for the moment
that they're not repeated.
320
00:19:43,620 --> 00:19:48,310
There, there they
were different, right?
321
00:19:48,310 --> 00:19:51,920
One and minus one in that, in
that, for that permutation.
322
00:19:51,920 --> 00:19:52,800
OK.
323
00:19:52,800 --> 00:19:59,460
So and after I found this
lambda, can I just look ahead?
324
00:19:59,460 --> 00:20:02,200
How I going to find x?
325
00:20:02,200 --> 00:20:06,670
After I have found this lambda,
the lambda being this --
326
00:20:06,670 --> 00:20:10,551
one of the numbers that
makes this matrix singular.
327
00:20:10,551 --> 00:20:11,550
Their product was eight.
328
00:20:11,550 --> 00:20:15,110
Then of course finding x
is just by elimination.
329
00:20:15,110 --> 00:20:15,610
Right?
330
00:20:15,610 --> 00:20:18,630
It's just -- now I've
got a singular matrix,
331
00:20:18,630 --> 00:20:20,640
I'm looking for the null space.
332
00:20:20,640 --> 00:20:24,020
We're experts at
finding the null space.
333
00:20:24,020 --> 00:20:26,570
You know, you do
elimination, you identify
334
00:20:26,570 --> 00:20:31,780
the, the, the pivot columns
and so on, you're --
335
00:20:31,780 --> 00:20:36,900
and, give values to
the free variables.
336
00:20:36,900 --> 00:20:39,300
Probably there'll only
be one free variable.
337
00:20:39,300 --> 00:20:42,480
We'll give it the
value one, like there.
338
00:20:42,480 --> 00:20:44,550
And we find the other variable.
339
00:20:44,550 --> 00:20:45,390
OK.
340
00:20:45,390 --> 00:20:52,610
So let's -- find the x
second will be a doable job.
341
00:20:52,610 --> 00:20:54,540
That's my big equation for x.
342
00:20:54,540 --> 00:20:58,070
Let's go, let's look at the
first job of finding lambda.
343
00:20:58,070 --> 00:20:59,500
Can I take another example?
344
00:20:59,500 --> 00:21:00,000
OK.
345
00:21:00,000 --> 00:21:02,130
And let's, let's
work that one out.
346
00:21:02,130 --> 00:21:02,630
OK.
347
00:21:02,630 --> 00:21:07,590
So let me take the example,
say, let me make it easy.
348
00:21:07,590 --> 00:21:09,240
it's just sitting there.
349
00:21:09,240 --> 00:21:12,550
Three three one and
one. what do you
350
00:21:12,550 --> 00:21:14,610
know about the complex numbers?
351
00:21:14,610 --> 00:21:16,680
So I've made it easy.
352
00:21:16,680 --> 00:21:19,160
I've made it two by two.
353
00:21:19,160 --> 00:21:20,810
I've made it symmetric.
354
00:21:20,810 --> 00:21:24,530
And I even made it
constant down the diagonal.
355
00:21:24,530 --> 00:21:27,830
That a matrix, a perfectly
real matrix could
356
00:21:27,830 --> 00:21:32,380
So that -- so the more, like,
special properties I stick
357
00:21:32,380 --> 00:21:36,270
into the matrix, the more
special outcome I get
358
00:21:36,270 --> 00:21:38,370
for the eigenvalues.
359
00:21:38,370 --> 00:21:42,050
For example, this
symmetric matrix,
360
00:21:42,050 --> 00:21:48,130
I know that it'll come out
with real eigenvalues. one.
361
00:21:48,130 --> 00:21:52,530
The eigenvalues will turn
out to be nice real numbers.
362
00:21:52,530 --> 00:21:57,760
And up in our previous example,
that was a symmetric matrix.
363
00:21:57,760 --> 00:22:02,130
Actually, while we're at it,
that was a symmetric matrix.
364
00:22:02,130 --> 00:22:05,500
Its eigenvalues were nice real
numbers, one and minus one.
365
00:22:05,500 --> 00:22:07,630
And do you notice anything
about its eigenvectors?
366
00:22:07,630 --> 00:22:08,970
And what do you notice?
367
00:22:08,970 --> 00:22:11,640
Anything particular about those
two vectors, one one and minus
368
00:22:11,640 --> 00:22:14,840
And now comes that thing that
I wanted to be reminded of.
369
00:22:14,840 --> 00:22:15,380
one one?
370
00:22:15,380 --> 00:22:18,850
They just happen to be -- no,
I can't say they just happen
371
00:22:18,850 --> 00:22:20,610
to be, because that's
the whole point,
372
00:22:20,610 --> 00:22:23,390
is that they had to be -- what?
373
00:22:23,390 --> 00:22:25,610
What are they?
374
00:22:25,610 --> 00:22:27,380
They're perpendicular.
375
00:22:27,380 --> 00:22:30,180
The vector, when I -- if I see
a vector one one and a one --
376
00:22:30,180 --> 00:22:33,121
and a minus one one, my
mind immediately takes that
377
00:22:33,121 --> 00:22:33,620
dot product.
378
00:22:33,620 --> 00:22:36,570
It's zero. what's the
determinant of that matrix?
379
00:22:36,570 --> 00:22:38,140
Those vectors are perpendicular.
380
00:22:38,140 --> 00:22:40,090
That'll happen here too.
381
00:22:40,090 --> 00:22:42,670
Well, let's find
the eigenvalues.
382
00:22:42,670 --> 00:22:45,770
Actually, oh, my
example's too easy.
383
00:22:45,770 --> 00:22:48,350
My example is too easy.
384
00:22:48,350 --> 00:22:53,550
Let me tell you in advance
what's going to happen.
385
00:22:53,550 --> 00:22:56,210
May I?
386
00:22:56,210 --> 00:22:58,510
Or shall I do the determinant
of A minus lambda,
387
00:22:58,510 --> 00:23:00,720
and then point out at the end?
388
00:23:00,720 --> 00:23:03,180
Will you remind me at
the -- after I've found
389
00:23:03,180 --> 00:23:10,830
the eigenvalues to say why they
were -- why they were easy from
390
00:23:10,830 --> 00:23:17,050
That -- it had to be eight,
because we factored into lambda
391
00:23:17,050 --> 00:23:20,160
the, from the example we did?
392
00:23:20,160 --> 00:23:23,270
OK, let's do the job here.
393
00:23:23,270 --> 00:23:27,410
Let's compute determinant
of A minus lambda I.
394
00:23:27,410 --> 00:23:29,490
So that's a determinant.
395
00:23:29,490 --> 00:23:32,600
And what's, what is this thing?
396
00:23:32,600 --> 00:23:36,090
It's the matrix A with lambda
removed from the diagonal.
397
00:23:36,090 --> 00:23:38,570
for this matrix?
398
00:23:38,570 --> 00:23:40,770
So the diagonal
matrix is shifted,
399
00:23:40,770 --> 00:23:43,800
and then I'm taking
the determinant.
400
00:23:43,800 --> 00:23:44,370
OK.
401
00:23:44,370 --> 00:23:47,390
So I multiply this out.
402
00:23:47,390 --> 00:23:49,340
So what is that determinant?
403
00:23:49,340 --> 00:23:52,520
Do you notice, I
didn't take lambda away
404
00:23:52,520 --> 00:23:55,090
from all the entries.
405
00:23:55,090 --> 00:23:56,790
It's lambda I, so
it's lambda along the
406
00:23:56,790 --> 00:23:58,040
Lambda plus three x. diagonal.
407
00:23:58,040 --> 00:24:04,040
So I get three minus lambda
squared and then minus one,
408
00:24:04,040 --> 00:24:06,380
right?
409
00:24:06,380 --> 00:24:08,000
And I want that to be zero.
410
00:24:08,000 --> 00:24:09,709
And what is A minus lambda I x?
411
00:24:09,709 --> 00:24:11,000
Well, I'm going to simplify it.
412
00:24:11,000 --> 00:24:11,050
And what will I get?
413
00:24:11,050 --> 00:24:11,190
So if I multiply this out, I
get lambda squared minus six
414
00:24:11,190 --> 00:24:11,290
What's -- how is this matrix
related to that matrix?
415
00:24:11,290 --> 00:24:11,330
lambda plus what?
416
00:24:11,330 --> 00:24:11,350
Plus eight.
417
00:24:11,350 --> 00:24:11,400
But it's out there.
418
00:24:11,400 --> 00:24:11,490
And that I'm going
to set to zero.
419
00:24:11,490 --> 00:24:11,560
And I'm going to solve it.
420
00:24:11,560 --> 00:24:11,640
So and it's, it's a
quadratic equation.
421
00:24:11,640 --> 00:24:11,750
I can use factorization, I
can use the quadratic formula.
422
00:24:11,750 --> 00:24:12,625
I'll get two lambdas.
423
00:24:12,625 --> 00:24:30,690
Before I do it, tell me
what's that number six that's
424
00:24:30,690 --> 00:24:41,560
showing up in this equation?
425
00:24:41,560 --> 00:24:48,080
It's the trace.
426
00:24:48,080 --> 00:25:03,300
That number six is
three plus three.
427
00:25:03,300 --> 00:25:05,820
And while we're at it, what's
the number eight that's
428
00:25:05,820 --> 00:25:08,590
showing up in this equation?
429
00:25:08,590 --> 00:25:10,340
It's the determinant.
430
00:25:10,340 --> 00:25:13,290
That our matrix has
determinant eight.
431
00:25:13,290 --> 00:25:16,820
So in a two by two
case, it's really nice.
432
00:25:16,820 --> 00:25:21,630
It's lambda squared minus
the trace times lambda --
433
00:25:21,630 --> 00:25:24,430
the trace is the
linear coefficient --
434
00:25:24,430 --> 00:25:27,880
and plus the determinant,
the constant term.
435
00:25:27,880 --> 00:25:28,540
OK.
436
00:25:28,540 --> 00:25:32,730
So let's -- can, can
we find the roots?
437
00:25:32,730 --> 00:25:36,310
I guess the easy way is to
factor that as something times
438
00:25:36,310 --> 00:25:37,980
something.
439
00:25:37,980 --> 00:25:41,280
If we couldn't factor it,
then we'd have to use the old
440
00:25:41,280 --> 00:25:46,590
b^2-4ac formula, but I, I think
we can factor that into lambda
441
00:25:46,590 --> 00:25:50,870
minus what times
lambda minus what?
442
00:25:50,870 --> 00:25:54,850
Can you do that factorization?
443
00:25:54,850 --> 00:25:57,240
Four and two?
444
00:25:57,240 --> 00:25:59,172
Lambda minus four
times lambda minus two.
445
00:25:59,172 --> 00:26:00,880
So the, the eigenvalues
are four and two.
446
00:26:00,880 --> 00:26:01,210
So the eigenvalues are --
one eigenvalue, lambda one,
447
00:26:01,210 --> 00:26:01,300
Now I'm looking for x, the
eigenvector. let's say,
448
00:26:01,300 --> 00:26:01,320
is four.
449
00:26:01,320 --> 00:26:01,490
Lambda two, the other
eigenvalue, is two.
450
00:26:01,490 --> 00:26:03,570
The eigenvalues
are four and two.
451
00:26:03,570 --> 00:26:08,540
And then I can go
for the eigenvectors.
452
00:26:08,540 --> 00:26:15,380
Suppose I have a matrix
A, and Ax equal lambda x.
453
00:26:15,380 --> 00:26:16,620
equals zero.
454
00:26:16,620 --> 00:26:20,980
You see I got the
eigenvalues first.
455
00:26:20,980 --> 00:26:25,950
So if they, if this
had eigenvalue lambda,
456
00:26:25,950 --> 00:26:26,860
Four and two.
457
00:26:26,860 --> 00:26:29,030
Now for the eigenvectors.
458
00:26:29,030 --> 00:26:31,300
So what are the eigenvectors?
459
00:26:31,300 --> 00:26:34,880
They're these guys in
the null space when
460
00:26:34,880 --> 00:26:40,090
I take away, when I make the
matrix singular by taking
461
00:26:40,090 --> 00:26:42,740
four I or two I away.
462
00:26:42,740 --> 00:26:46,470
So we're -- we got to
do those separately.
463
00:26:46,470 --> 00:26:50,100
I'll -- let me find the
eigenvector for four first.
464
00:26:50,100 --> 00:26:57,420
So I'll subtract four,
so A minus four I is --
465
00:26:57,420 --> 00:27:00,590
so taking four away will
put minus ones there.
466
00:27:04,070 --> 00:27:07,000
And what's the point
about that matrix?
467
00:27:07,000 --> 00:27:10,260
If four is an eigenvalue,
then A minus four
468
00:27:10,260 --> 00:27:13,220
I had better be a
what kind of matrix?
469
00:27:13,220 --> 00:27:14,800
Singular.
470
00:27:14,800 --> 00:27:17,800
If that matrix isn't singular,
the four wasn't correct.
471
00:27:17,800 --> 00:27:21,280
But we're OK, that
matrix is singular.
472
00:27:21,280 --> 00:27:23,110
And what's the x now?
473
00:27:23,110 --> 00:27:25,290
The x is in the null space.
474
00:27:25,290 --> 00:27:28,910
So what's the x1 that goes
with, with the lambda one?
475
00:27:28,910 --> 00:27:32,580
eigenvalue, eigenvector,
eigenvalue for this,
476
00:27:32,580 --> 00:27:38,294
So that A -- so this is -- now
I'm doing A x1 is lambda one
477
00:27:38,294 --> 00:27:38,794
x1.
478
00:27:41,640 --> 00:27:45,160
So I took A minus lambda
one I, that's this matrix,
479
00:27:45,160 --> 00:27:48,630
and now I'm looking for
the x1 in its null space,
480
00:27:48,630 --> 00:27:49,510
and who is he?
481
00:27:49,510 --> 00:27:51,760
What's the vector x
in the null space?
482
00:27:51,760 --> 00:27:53,380
Of course it's one one.
483
00:27:53,380 --> 00:27:56,310
So that's the eigenvector that
goes with that eigenvalue.
484
00:27:56,310 --> 00:27:57,610
So, so now --
485
00:27:57,610 --> 00:28:00,540
Let's just spend one
more minute on this bad
486
00:28:00,540 --> 00:28:02,490
Now how about the
eigenvector that
487
00:28:02,490 --> 00:28:04,110
goes with the other eigenvalue?
488
00:28:04,110 --> 00:28:06,040
Can I do that
with, with erasing?
489
00:28:06,040 --> 00:28:08,940
I take A minus two I.
490
00:28:08,940 --> 00:28:11,380
So now I take two away
from the diagonal,
491
00:28:11,380 --> 00:28:15,670
and that leaves me
with a one and a one.
492
00:28:15,670 --> 00:28:19,490
So A minus two I has again
produced a singular matrix,
493
00:28:19,490 --> 00:28:21,710
as it had to.
494
00:28:21,710 --> 00:28:24,990
I'm looking for the
null space of that guy.
495
00:28:24,990 --> 00:28:27,210
What vector is in
its null space?
496
00:28:27,210 --> 00:28:29,320
Well, of course, a
whole line of vectors.
497
00:28:32,040 --> 00:28:35,990
So when I say the eigenvector,
I'm not speaking correctly.
498
00:28:35,990 --> 00:28:38,490
There's a whole line of
eigenvectors, and you just --
499
00:28:38,490 --> 00:28:40,760
I just want a basis.
500
00:28:40,760 --> 00:28:43,360
And for a line I
just want one vector.
501
00:28:43,360 --> 00:28:48,130
But -- You could, you're
-- there's some freedom
502
00:28:48,130 --> 00:28:50,860
in choosing that one, but
choose a reasonable one.
503
00:28:50,860 --> 00:28:54,200
What's a vector in the
null space of that?
504
00:28:54,200 --> 00:28:58,800
Well, the natural vector
to pick as the eigenvector
505
00:28:58,800 --> 00:29:01,540
with, with lambda
two is minus one one.
506
00:29:05,130 --> 00:29:07,760
If I did elimination
on that vector
507
00:29:07,760 --> 00:29:10,520
and set that, the free
variable to be one,
508
00:29:10,520 --> 00:29:15,010
I would get minus one
and get that eigenvector.
509
00:29:15,010 --> 00:29:22,250
So you see then that
I've got eigenvector,
510
00:29:22,250 --> 00:29:32,890
Now the other neat fact
is that the determinant,
511
00:29:32,890 --> 00:29:36,906
How are those two
matrices related?
512
00:29:40,530 --> 00:29:46,710
Well, one is just three I more
than the other one, right? two.
513
00:29:46,710 --> 00:29:52,410
I just took that matrix and I --
514
00:29:52,410 --> 00:30:29,276
I took this matrix
and I added three I.
515
00:30:29,276 --> 00:30:31,900
So my question is, what happened
to the minus four times lambda
516
00:30:31,900 --> 00:30:34,720
minus two. eigenvalues and what
happened to the eigenvectors?
517
00:30:34,720 --> 00:30:37,530
That's the, that's like the
question we keep asking now
518
00:30:37,530 --> 00:30:39,500
in this chapter.
519
00:30:39,500 --> 00:30:42,690
If I, if I do something to the
matrix, what happens if I --
520
00:30:42,690 --> 00:30:44,730
or I know something
about the matrix,
521
00:30:44,730 --> 00:30:48,550
what's the what's the
conclusion for its eigenvectors
522
00:30:48,550 --> 00:30:49,280
and eigenvalues?
523
00:30:49,280 --> 00:30:54,180
Because -- those eigenvalues and
eigenvectors are going to tell
524
00:30:54,180 --> 00:30:57,280
us important information
about the matrix.
525
00:30:57,280 --> 00:31:00,970
And here what are we seeing?
526
00:31:00,970 --> 00:31:04,330
What's happening to these
eigenvalues, one and minus
527
00:31:04,330 --> 00:31:06,460
one, when I add three I?
528
00:31:10,040 --> 00:31:13,160
It just added three
to the eigenvalues.
529
00:31:13,160 --> 00:31:16,640
I got four and two, three
more than one and minus
530
00:31:16,640 --> 00:31:18,220
one.
531
00:31:18,220 --> 00:31:19,960
What happened to
the eigenvectors?
532
00:31:19,960 --> 00:31:21,490
Nothing at all.
533
00:31:21,490 --> 00:31:25,170
One one is -- and minus -- and
one -- and minus one one are --
534
00:31:25,170 --> 00:31:27,050
is still the eigenvectors.
535
00:31:27,050 --> 00:31:34,200
In other words, simple
but useful observation.
536
00:31:34,200 --> 00:31:40,320
If I add three I to a matrix,
its eigenvectors don't change
537
00:31:40,320 --> 00:31:42,870
and its eigenvalues
are three bigger.
538
00:31:42,870 --> 00:31:44,130
Let's, let's just see why.
539
00:31:44,130 --> 00:32:02,150
Let me keep all this on the same
board. but just so you see --
540
00:32:02,150 --> 00:32:08,410
so I'll try to do that.
541
00:32:08,410 --> 00:32:39,970
this has eigenvalue
lambda plus three.
542
00:32:39,970 --> 00:32:46,590
And x, the eigenvector, is
the same x for both matrices.
543
00:32:46,590 --> 00:32:48,790
OK.
544
00:32:48,790 --> 00:32:50,195
So that's, great.
545
00:32:53,440 --> 00:32:54,670
Of course, it's special.
546
00:32:54,670 --> 00:32:57,570
We got the new matrix
by adding three I.
547
00:32:57,570 --> 00:32:59,500
Suppose I had added
another matrix.
548
00:32:59,500 --> 00:33:02,410
Suppose I know the eigenvalues
and eigenvectors of A.
549
00:33:02,410 --> 00:33:07,250
So I took A minus lambda I x,
and what kind of a matrix I
550
00:33:07,250 --> 00:33:09,510
So this is, this,
this little board
551
00:33:09,510 --> 00:33:11,875
here is going to
be not so great.
552
00:33:17,730 --> 00:33:21,270
Suppose I have a matrix A and
it has an eigenvector x with
553
00:33:21,270 --> 00:33:23,260
an eigenvalue lambda.
554
00:33:23,260 --> 00:33:28,730
You remember, I solve
A minus lambda I x
555
00:33:28,730 --> 00:33:31,020
And now I add on
some other matrix.
556
00:33:31,020 --> 00:33:34,000
So, so what I'm asking you is,
if you know the eigenvalues
557
00:33:34,000 --> 00:33:38,430
of A and you know
the eigenvalues of B,
558
00:33:38,430 --> 00:33:43,990
let me say suppose B -- so this
is if -- let me put an if here.
559
00:33:43,990 --> 00:33:49,230
If Ax equals lambda x, fine,
and B has, eigenvalues,
560
00:33:49,230 --> 00:33:52,240
has eigenvalues --
561
00:33:52,240 --> 00:33:56,800
what shall we call them?
562
00:33:56,800 --> 00:34:02,270
Alpha, alpha one and alpha --
563
00:34:02,270 --> 00:34:05,020
let's say --
564
00:34:05,020 --> 00:34:07,270
I'll use alpha for
the eigenvalues of B
565
00:34:07,270 --> 00:34:08,199
for no good reason.
566
00:34:11,489 --> 00:34:16,389
What a- you see what I'm going
to ask is, how about A plus B?
567
00:34:20,949 --> 00:34:24,070
Let me, let me give you
the, let me give you,
568
00:34:24,070 --> 00:34:27,770
what you might think first.
569
00:34:27,770 --> 00:34:28,730
OK.
570
00:34:28,730 --> 00:34:35,460
If Ax equals lambda x and if
B has an eigenvalue alpha,
571
00:34:35,460 --> 00:34:40,630
then I allowed to say -- what's
the matter with this argument?
572
00:34:40,630 --> 00:34:43,300
That gave us the
constant term eight.
573
00:34:43,300 --> 00:34:44,070
It's wrong.
574
00:34:44,070 --> 00:34:47,130
What I'm going to
write up is wrong.
575
00:34:47,130 --> 00:34:50,179
I'm going to say Bx is alpha x.
576
00:34:50,179 --> 00:34:55,909
Add those up, and you get A plus
B x equals lambda plus alpha x.
577
00:34:55,909 --> 00:35:00,500
So you would think that if
you know the eigenvalues of A
578
00:35:00,500 --> 00:35:03,490
and you knew the
eigenvalues of B,
579
00:35:03,490 --> 00:35:07,900
then if you added you would know
the eigenvalues of A plus B.
580
00:35:07,900 --> 00:35:11,310
But that's false.
581
00:35:11,310 --> 00:35:17,840
A plus B -- well, when B was
three I, that worked great.
582
00:35:17,840 --> 00:35:21,290
But this is not so great.
583
00:35:21,290 --> 00:35:25,720
And what's the matter
with that argument there?
584
00:35:25,720 --> 00:35:32,610
We have no reason to
believe that x is also
585
00:35:32,610 --> 00:35:33,500
an eigenvector of
586
00:35:33,500 --> 00:35:38,930
B has some eigenvalues,
B. but it's
587
00:35:38,930 --> 00:35:43,600
got some different
eigenvectors normally.
588
00:35:43,600 --> 00:35:45,520
It's a different matrix.
589
00:35:45,520 --> 00:35:47,250
I don't know anything special.
590
00:35:47,250 --> 00:35:50,150
If I don't know anything
special, then as far as I know,
591
00:35:50,150 --> 00:35:52,670
it's got some different
eigenvector y,
592
00:35:52,670 --> 00:35:55,790
and when I add I
get just rubbish.
593
00:35:55,790 --> 00:35:57,480
I mean, I get --
594
00:35:57,480 --> 00:35:59,970
I can add, but I
don't learn anything.
595
00:35:59,970 --> 00:36:05,380
So not so great is A plus B.
596
00:36:05,380 --> 00:36:09,460
Or A times B.
597
00:36:09,460 --> 00:36:12,670
Normally the
eigenvalues of A plus B
598
00:36:12,670 --> 00:36:18,630
or A times B are not eigenvalues
of A plus eigenvalues of B.
599
00:36:18,630 --> 00:36:22,500
Ei- eigenvalues are
not, like, linear.
600
00:36:22,500 --> 00:36:24,610
Or -- and they don't multiply.
601
00:36:24,610 --> 00:36:27,580
Because, eigenvectors
are usually different
602
00:36:27,580 --> 00:36:31,600
and, and there's just
no way to find out
603
00:36:31,600 --> 00:36:33,520
what A plus B does to affect
604
00:36:33,520 --> 00:36:34,960
What do I do now? it.
605
00:36:34,960 --> 00:36:35,470
OK.
606
00:36:35,470 --> 00:36:41,660
So that's, like, a caution.
607
00:36:41,660 --> 00:36:44,610
Don't, if B is a multiple
of the identity, great,
608
00:36:44,610 --> 00:36:50,340
but if B is some general matrix,
then for A plus B you've got
609
00:36:50,340 --> 00:36:54,580
to find -- you've got to
solve the eigenvalue problem.
610
00:36:54,580 --> 00:36:59,720
Now I want to do another
example that brings out a,
611
00:36:59,720 --> 00:37:02,200
OK. another point
about eigenvalues.
612
00:37:02,200 --> 00:37:06,280
Let me make this example
a rotation matrix.
613
00:37:06,280 --> 00:37:09,390
possibility of complex numbers.
614
00:37:09,390 --> 00:37:10,170
OK.
615
00:37:10,170 --> 00:37:13,280
So here's another example.
616
00:37:13,280 --> 00:37:16,390
So a rotate --
617
00:37:16,390 --> 00:37:21,060
oh, I'd better call it Q.
618
00:37:21,060 --> 00:37:26,500
I often use Q for,
for, rotations
619
00:37:26,500 --> 00:37:32,600
because those are the, like,
very important examples
620
00:37:32,600 --> 00:37:34,580
of orthogonal matrices.
621
00:37:34,580 --> 00:37:38,290
Let me make it a
ninety degree rotation.
622
00:37:38,290 --> 00:37:41,709
So -- my matrix is going to
be the one that rotates every
623
00:37:41,709 --> 00:37:43,750
And that's the sum, that's
lambda one plus lambda
624
00:37:43,750 --> 00:37:46,640
vector by ninety degrees.
625
00:37:46,640 --> 00:37:49,480
So do you remember that matrix?
626
00:37:49,480 --> 00:37:51,950
It's the cosine of
ninety degrees, which
627
00:37:51,950 --> 00:37:54,790
is zero, the sine
of ninety degrees,
628
00:37:54,790 --> 00:38:03,990
which is one, minus the sine of
ninety, the cosine of ninety.
629
00:38:03,990 --> 00:38:09,360
So that matrix
deserves the letter Q.
630
00:38:09,360 --> 00:38:15,500
It's an orthogonal matrix,
very, very orthogonal matrix.
631
00:38:15,500 --> 00:38:21,420
Now I'm interested in its
eigenvalues and eigenvectors.
632
00:38:21,420 --> 00:38:24,500
Two by two, it
can't be that tough.
633
00:38:24,500 --> 00:38:27,200
We know that the
eigenvalues add to zero.
634
00:38:30,780 --> 00:38:33,310
Actually, we know
something already here.
635
00:38:33,310 --> 00:38:35,750
The eigen- what's the sum
of the two eigenvalues?
636
00:38:35,750 --> 00:38:38,880
Just tell me what I just said.
637
00:38:38,880 --> 00:38:40,430
Zero, right.
638
00:38:40,430 --> 00:38:42,310
From that trace business.
639
00:38:42,310 --> 00:38:46,550
The sum of the eigenvalues
is, is going to come out zero.
640
00:38:46,550 --> 00:38:48,440
And the product of
the eigenvalues,
641
00:38:48,440 --> 00:38:50,190
did I tell you about
the determinant being
642
00:38:50,190 --> 00:38:50,870
the product of the eigenvalues?
643
00:38:50,870 --> 00:38:50,880
No.
644
00:38:50,880 --> 00:38:50,950
But that's a good thing to know.
645
00:38:50,950 --> 00:38:51,030
We pointed out how
that eight appeared in
646
00:38:51,030 --> 00:38:52,863
the, in the quadratic
equation. eigenvalues,
647
00:38:52,863 --> 00:39:07,700
we can postpone that evil day,
648
00:39:07,700 --> 00:39:23,700
So let me just say this.
649
00:39:23,700 --> 00:39:49,930
The trace is zero
plus zero, obviously.
650
00:39:49,930 --> 00:39:52,675
And that was the determinant.
651
00:39:52,675 --> 00:39:53,175
OK.
652
00:39:56,230 --> 00:39:58,390
What I'm leading up
to with this example
653
00:39:58,390 --> 00:40:03,610
is that something's
going to go wrong.
654
00:40:03,610 --> 00:40:09,080
Something goes
wrong for rotation
655
00:40:09,080 --> 00:40:16,410
because what vector can
come out parallel to itself
656
00:40:16,410 --> 00:40:18,820
after a rotation?
657
00:40:18,820 --> 00:40:23,790
If this matrix rotates every
vector by ninety degrees,
658
00:40:23,790 --> 00:40:26,880
what could be an eigenvector?
659
00:40:26,880 --> 00:40:31,190
Do you see we're, we're,
we're going to have trouble.
660
00:40:31,190 --> 00:40:33,780
eigenvectors are --
661
00:40:36,060 --> 00:40:36,560
Well.
662
00:40:36,560 --> 00:40:39,220
Our, our picture
of eigenvectors,
663
00:40:39,220 --> 00:40:42,410
of, of coming out in the same
direction that they went in,
664
00:40:42,410 --> 00:40:45,140
there won't be it.
665
00:40:45,140 --> 00:40:48,820
And with, and with eigenvalues
we're going to have trouble.
666
00:40:48,820 --> 00:40:50,440
From these equations.
667
00:40:50,440 --> 00:40:51,710
Let's see.
668
00:40:51,710 --> 00:40:53,780
Why I expecting trouble?
669
00:40:53,780 --> 00:40:56,640
The, the first equation
says that the eigenvalues
670
00:40:56,640 --> 00:40:57,320
add to zero.
671
00:40:59,920 --> 00:41:01,170
So there's a plus and a minus.
672
00:41:01,170 --> 00:41:02,211
So I take the eigenvalue.
673
00:41:04,360 --> 00:41:06,490
But then the second
equation says
674
00:41:06,490 --> 00:41:09,450
that the product is plus one.
675
00:41:09,450 --> 00:41:10,880
We're in trouble.
676
00:41:10,880 --> 00:41:14,810
But there's a way out.
677
00:41:14,810 --> 00:41:17,860
So how -- let's do
the usual stuff.
678
00:41:17,860 --> 00:41:21,700
Look at determinant
of Q minus lambda I.
679
00:41:21,700 --> 00:41:27,230
So I'll just follow the
rules, take the determinant,
680
00:41:27,230 --> 00:41:31,940
subtract lambda from the
diagonal, where I had zeros,
681
00:41:31,940 --> 00:41:34,400
the rest is the same.
682
00:41:34,400 --> 00:41:37,180
Rest of Q is just copied.
683
00:41:37,180 --> 00:41:38,540
Compute that determinant.
684
00:41:38,540 --> 00:41:42,720
OK, so what does that
determinant equal?
685
00:41:42,720 --> 00:41:47,740
Lambda squared minus
minus one plus what?
686
00:41:51,930 --> 00:41:54,060
What's up?
687
00:41:54,060 --> 00:41:56,020
There's my equation.
688
00:41:56,020 --> 00:41:59,050
My equation for the
eigenvalues is lambda
689
00:41:59,050 --> 00:42:00,950
squared plus one equals zero.
690
00:42:00,950 --> 00:42:04,620
What are the eigenvalues
lambda one and lambda two?
691
00:42:04,620 --> 00:42:27,340
They're I, whatever that
is, and minus it, right.
692
00:42:27,340 --> 00:42:30,240
Those are the right numbers.
693
00:42:30,240 --> 00:42:36,270
To be real numbers even though
the matrix was perfectly real.
694
00:42:36,270 --> 00:42:38,190
So this can happen.
695
00:42:41,060 --> 00:42:44,950
Complex numbers are going to --
have to enter eighteen oh six
696
00:42:44,950 --> 00:42:51,500
at this moment.
697
00:42:51,500 --> 00:42:54,600
Boo, right.
698
00:42:54,600 --> 00:42:56,950
All right.
699
00:42:56,950 --> 00:43:02,970
If I just choose good
matrices that have real
700
00:43:02,970 --> 00:43:08,500
supposed to have here?
701
00:43:24,360 --> 00:43:35,590
We do know a little
information about the,
702
00:43:35,590 --> 00:43:38,672
the two complex numbers.
703
00:43:38,672 --> 00:43:40,380
They're complex
conjugates of each other.
704
00:43:45,850 --> 00:43:51,590
If, if lambda is an
eigenvalue, then when I change,
705
00:43:51,590 --> 00:43:54,360
when I go -- you remember
what complex conjugates are?
706
00:43:54,360 --> 00:43:57,170
You switch the sign
of the imaginary part.
707
00:43:57,170 --> 00:43:59,910
Well, this was only
imaginary, had no real part,
708
00:43:59,910 --> 00:44:02,690
so we just switched its sign.
709
00:44:02,690 --> 00:44:06,720
So that eigenvalues
come in pairs like that,
710
00:44:06,720 --> 00:44:08,440
but they're complex.
711
00:44:08,440 --> 00:44:11,040
A complex conjugate pair.
712
00:44:11,040 --> 00:44:14,530
And that can happen with
a perfectly real matrix.
713
00:44:14,530 --> 00:44:17,060
And as a matter of fact --
714
00:44:17,060 --> 00:44:18,930
so that was my,
my point earlier,
715
00:44:18,930 --> 00:44:23,240
that if a matrix was
symmetric, it wouldn't happen.
716
00:44:23,240 --> 00:44:27,090
So if we stick to matrices that
are symmetric or, like, close
717
00:44:27,090 --> 00:44:32,610
to symmetric, then the
eigenvalues will stay real.
718
00:44:32,610 --> 00:44:35,400
But if we move far
away from symmetric --
719
00:44:35,400 --> 00:44:39,520
and that's as far as you can
move, because that matrix is --
720
00:44:39,520 --> 00:44:44,470
how is Q transpose related
to Q for that matrix?
721
00:44:44,470 --> 00:44:46,760
That matrix is anti-symmetric.
722
00:44:46,760 --> 00:44:49,520
Q transpose is minus Q.
723
00:44:49,520 --> 00:44:52,070
That's the very
opposite of symmetry.
724
00:44:52,070 --> 00:44:54,570
When I flip across
the diagonal I get --
725
00:44:54,570 --> 00:44:56,350
I reverse all the signs.
726
00:44:56,350 --> 00:45:00,840
Those are the guys that have
pure imaginary eigenvalues.
727
00:45:00,840 --> 00:45:03,320
So they're the extreme case.
728
00:45:03,320 --> 00:45:07,380
And in between are,
are matrices that
729
00:45:07,380 --> 00:45:10,990
are not symmetric or
anti-symmetric but,
730
00:45:10,990 --> 00:45:13,710
but they have partly
a symmetric part
731
00:45:13,710 --> 00:45:15,110
and an anti-symmetric part.
732
00:45:15,110 --> 00:45:16,660
OK.
733
00:45:16,660 --> 00:45:23,220
So I'm doing a bunch of examples
here to show the possibilities.
734
00:45:23,220 --> 00:45:28,320
The good possibilities being
perpendicular eigenvectors,
735
00:45:28,320 --> 00:45:30,360
real eigenvalues.
736
00:45:30,360 --> 00:45:33,910
The bad possibilities
being complex eigenvalues.
737
00:45:33,910 --> 00:45:37,360
We could say that's bad.
738
00:45:37,360 --> 00:45:39,455
There's another even worse.
739
00:45:42,380 --> 00:45:45,740
I'm getting through the,
the bad things here today.
740
00:45:45,740 --> 00:45:51,750
Then, then the next
lecture can, can,
741
00:45:51,750 --> 00:45:56,750
can be like pure happiness.
742
00:45:56,750 --> 00:45:57,720
OK.
743
00:45:57,720 --> 00:46:03,260
Here's one more bad
thing that could happen.
744
00:46:03,260 --> 00:46:05,940
So I, again, I'll do
it with an example.
745
00:46:05,940 --> 00:46:10,740
Suppose my matrix is, suppose
I take this three three one
746
00:46:10,740 --> 00:46:13,140
and I change that guy to zero.
747
00:46:18,150 --> 00:46:21,280
What are the eigenvalues
of that matrix?
748
00:46:21,280 --> 00:46:22,740
What are the eigenvectors?
749
00:46:22,740 --> 00:46:25,380
This is always our question.
750
00:46:25,380 --> 00:46:26,860
Of course, the
next section we're
751
00:46:26,860 --> 00:46:29,580
going to show why
are, why do we care.
752
00:46:29,580 --> 00:46:33,121
But for the moment, this
lecture is introducing
753
00:46:33,121 --> 00:46:33,620
them.
754
00:46:33,620 --> 00:46:35,990
And let's just find them.
755
00:46:35,990 --> 00:46:36,640
OK.
756
00:46:36,640 --> 00:46:40,530
What are the eigenvalues
of that matrix?
757
00:46:40,530 --> 00:46:45,910
Let me tell you -- at a glance
we could answer that question.
758
00:46:45,910 --> 00:46:49,490
Because the matrix
is triangular.
759
00:46:49,490 --> 00:46:53,540
It's really useful to know --
if you've got properties like
760
00:46:53,540 --> 00:46:55,320
a triangular matrix.
761
00:46:55,320 --> 00:46:57,980
It's very useful to know
you can read the eigenvalues
762
00:46:57,980 --> 00:46:58,610
off.
763
00:46:58,610 --> 00:47:01,920
They're right on the diagonal.
764
00:47:01,920 --> 00:47:05,470
So the eigenvalue is
three and also three.
765
00:47:05,470 --> 00:47:07,290
Three is a repeated eigenvalue.
766
00:47:07,290 --> 00:47:09,330
But let's see that happen.
767
00:47:09,330 --> 00:47:10,660
Let me do it right.
768
00:47:10,660 --> 00:47:15,370
The determinant of A minus
lambda I, what I always
769
00:47:15,370 --> 00:47:17,119
have to do is this determinant.
770
00:47:17,119 --> 00:47:18,660
I take away lambda
from the diagonal.
771
00:47:21,600 --> 00:47:24,070
I leave the rest.
772
00:47:24,070 --> 00:47:28,450
I compute the determinant,
so I get a three minus lambda
773
00:47:28,450 --> 00:47:32,090
times a three minus lambda.
774
00:47:32,090 --> 00:47:35,680
And nothing.
775
00:47:35,680 --> 00:47:38,860
So that's where the
triangular part came in.
776
00:47:38,860 --> 00:47:41,260
Triangular part, the one
thing we know about triangular
777
00:47:41,260 --> 00:47:44,660
matrices is the determinant
is just the product down
778
00:47:44,660 --> 00:47:45,890
the diagonal.
779
00:47:45,890 --> 00:47:48,990
And in this case, it's
this same, repeated --
780
00:47:48,990 --> 00:47:51,775
so lambda one is one --
781
00:47:51,775 --> 00:47:53,900
sorry, lambda one is three
and lambda two is three.
782
00:47:53,900 --> 00:47:58,540
That was easy.
783
00:47:58,540 --> 00:48:05,900
I mean, no -- why should I
be pessimistic about a matrix
784
00:48:05,900 --> 00:48:10,650
whose eigenvalues can
be read off right away?
785
00:48:10,650 --> 00:48:14,820
The problem with this matrix
is in the eigenvectors.
786
00:48:14,820 --> 00:48:16,450
So let's go to the eigenvectors.
787
00:48:16,450 --> 00:48:18,960
So how do I find
the eigenvectors?
788
00:48:18,960 --> 00:48:22,010
I'm looking for a
couple of eigenvectors.
789
00:48:22,010 --> 00:48:22,800
Singular, right?
790
00:48:22,800 --> 00:48:31,590
It's supposed to be singular.
791
00:48:31,590 --> 00:48:40,530
And then it's got some
vectors -- which it is.
792
00:48:40,530 --> 00:48:59,410
So it's got some vector
x in the null space.
793
00:49:12,890 --> 00:49:15,670
And what, what's the, what's
-- give me a basis for the null
794
00:49:15,670 --> 00:49:18,360
space for that guy.
795
00:49:18,360 --> 00:49:21,860
Tell me, what's a vector x
in the null space, so that'll
796
00:49:21,860 --> 00:49:25,710
be the, the eigenvector
that goes with lambda one
797
00:49:25,710 --> 00:49:27,030
equals three.
798
00:49:27,030 --> 00:49:31,030
The eigenvector is -- so
what's in the null space?
799
00:49:31,030 --> 00:49:32,180
One zero, is it?
800
00:49:34,700 --> 00:49:35,200
Great.
801
00:49:38,730 --> 00:49:40,600
Now, what's the
other eigenvector?
802
00:49:40,600 --> 00:49:47,960
What's, what's the eigenvector
that goes with lambda two?
803
00:49:47,960 --> 00:49:51,620
Well, lambda two is three again.
804
00:49:51,620 --> 00:49:53,150
So I get the same thing again.
805
00:49:53,150 --> 00:49:55,500
Give me another vector --
806
00:49:55,500 --> 00:49:57,750
I want it to be independent.
807
00:49:57,750 --> 00:49:59,340
If I'm going to
write down an x2,
808
00:49:59,340 --> 00:50:02,440
I'm never going to let
it be dependent on x1.
809
00:50:02,440 --> 00:50:05,240
I'm looking for
independent eigenvectors,
810
00:50:05,240 --> 00:50:08,720
and what's the conclusion?
811
00:50:08,720 --> 00:50:11,010
There isn't one.
812
00:50:11,010 --> 00:50:17,050
This is a degenerate matrix.
813
00:50:17,050 --> 00:50:22,960
It's only got one line of
eigenvectors instead of two.
814
00:50:22,960 --> 00:50:27,250
It's this possibility
of a repeated eigenvalue
815
00:50:27,250 --> 00:50:34,390
opens this further possibility
of a shortage of eigenvectors.
816
00:50:34,390 --> 00:50:43,310
And so there's no second
independent eigenvector x2.
817
00:50:43,310 --> 00:50:48,010
So it's a matrix, it's
a two by two matrix,
818
00:50:48,010 --> 00:50:51,850
but with only one
independent eigenvector.
819
00:50:51,850 --> 00:50:56,910
So that will be --
the matrices that --
820
00:50:56,910 --> 00:51:01,830
where eigenvectors are --
don't give the complete story.
821
00:51:01,830 --> 00:51:02,330
OK.
822
00:51:02,330 --> 00:51:05,930
My lecture on Monday will
give the complete story
823
00:51:05,930 --> 00:51:11,290
for all the other matrices.
824
00:51:11,290 --> 00:51:12,360
Thanks.
825
00:51:12,360 --> 00:51:16,640
Have a good weekend.
826
00:51:16,640 --> 00:51:22,480
A real New England weekend.