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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 most of the rest of

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the course.

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It's, again,
matrices are square and we're

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looking now for some special
numbers,

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the eigenvalues,
and some special vectors,

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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 comes

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a vector Ax.

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It's like a function.

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With a function in calculus,
in goes a number x,

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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 are the ones the

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come out in the same direction
that they went in.

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That won't be typical.

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Most vectors,
Ax is in -- points in

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some different direction.

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But there are certain vectors
where Ax comes out parallel to

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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 won't

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be eigenvectors -- that -- for
which Ax is in the same

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direction as x,
and by same direction I allow

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it to
be the very opposite direction,

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I allow lambda to be negative
or zero.

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Well, I guess we've met the
eigenvectors that have

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eigenvalue zero.

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Those are in the same
direction, but they're -- in a

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kind of very special way.

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So this --
the eigenvector x.

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Lambda, whatever this
multiplying factor is,

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whether it's six or minus six
or zero or even some imaginary

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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, that would mean that Ax

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was zero x, in other words zero.

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So what would x,
where would we look for -- what

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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

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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 -- will be the

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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

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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 multiplied

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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

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find them.

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But before I,
before I

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do that, and that's where
determinant will come in,

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can I just 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 -- what I've

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called A, now I'm going to call
it P for the

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moment, because it's -- I'm
thinking OK, let's look at a

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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,

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and the lambdas,
the eigenvalues,

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for -- and now let me say a
projection matrix.

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My,
my point is that we -- before

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we get into determinants and,
and formulas and all that

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stuff, let's take some matrices
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 c- because Pb,

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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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In this picture,
where could I start with a

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vector b or x,
do its projection,

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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 -- so any vector,

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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 -- I
called it b here,

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because this is our familiar
picture, but now I'm going to

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say that b was no good for,
for the, 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.

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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,

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which is -- just compare it
with that.

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The eigenvalue,
the multiplier,

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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, because I would like to get

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three, if I'm in three
dimensions, I would like to hope

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for three independent
eigenvectors,

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two 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

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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.

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Any x that's perpendicular to
the plane, what's Px for that,

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for that, vector?

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What's the projection of this
guy

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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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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,

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A inverse, A transpose thing,
but without doing that

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from the picture 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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Can you tell me a vector x --
see, we'll have a system soon

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enough, so I,
I would like to just do these

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e- these couple of examples,
just to see the picture before

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we, before we let it all,
go into a system where

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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 in the

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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,

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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 -- permuted turn out

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to be a multiple of x1 x2,
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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has eigenvalue one?

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So what, what vector would have
eigenvalue one,

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so that if I,
if I permute it it doesn't

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change?

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There, that could be one one,
thanks.

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One one.

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OK, take that vector one one.

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That will be an eigenvector,
because if I do Ax I get one

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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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But I have here a two by two
matrix, and I figure there's

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going to be a second eigenvalue.

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And eigenvector.

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Now, what about that?

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What's a vector,
OK, maybe we can just,

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like, guess it.

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A vector that the other --
actually, this one that I'm

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thinking of is going to be a
vector that has eigenvalue minus

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one.

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That's going to be my other
eigenvalue for this matrix.

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It's a -- notice the nice
positive or not negative matrix,

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but an eigenvalue is going to
come out negative.

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And can you guess,
spot the x that will work for

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that?

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So I want a,
a vector.

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When I multiply by A,
which reverses the two

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components, I want the thing to
come out minus the original.

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So what shall I send in in that
case?

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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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Lambda is minus one.

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Ax -- so Ax was x there and Ax
is minus x here.

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Can I just mention,
like, jump ahead,

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and point out a special little
fact about eigenvalues.

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n by n matrices will have n
eigenvalues.

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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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We'd have a third degree or a
fourth degree or an n-th degree

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equation.

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But here's one nice fact.

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There, there's one pleasant
fact.

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That the sum of the eigenvalues
equals the sum down the

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diagonal.

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That's called the trace,
and I put that in the lecture

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content specifically.

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So this is a neat fact,
the fact that sthe sum of the

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lambdas, add up the lambdas,
equals the sum -- what would

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you like me to,
shall I write that down?

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What I'm want to say in words
is the sum down

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the diagonal of A.

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Shall I write a11+a22+...+ ann.

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That's add up the diagonal
entries.

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In this example,
it's zero.

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In other words,
once I found this eigenvalue of

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one, I knew the other one had to
be minus

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one in this two by two case,
because in the two by two case,

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which is a good one to,
to, play with,

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the trace tells you right away
what the other eigenvalue is.

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So if I tell you one
eigenvalue, you can tell me the

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other one.

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We'll,
we'll have that -- we'll,

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we'll see that again.

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OK.

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Now can I -- I could give more
examples, but maybe it's time to

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face the, the equation,
Ax equal lambda x,

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and figure how are we going to
find x and lambda.

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OK.

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So this, so the question now is
how to find eigenvalues and

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eigenvectors.

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How to solve,
how to solve Ax equal lambda x

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when we've got two unknowns both
in the equation.

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OK.

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Here's the trick.

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Simple idea.

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Bring this onto the
same side.

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Rewrite.

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Bring this over as A minus
lambda times the identity x

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equals zero.

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Right?

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I have Ax minus lambda x,
so I brought that over and I've

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00:17:09 --> 00:17:14
got zero left on the,
on the right-hand side.

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OK.

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I don't know lambda and I don't
know x, but I do know something

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here.

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What I know is if I,
if I'm going to be able to

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solve this thing,
for some x that's not the zero

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vector, that's not,
that's a useless eigenvector,

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doesn't count.

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What I know now is that
this matrix must be what?

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If I'm going to be -- if there
is an x -- I don't -- right now

286
00:17:47 --> 00:17:49
I don't know what it is.

287
00:17:49 --> 00:17:53
I'm going to find lambda first,
actually.

288
00:17:53 --> 00:17:59
And -- but if there is an x,
it tells me that this matrix,

289
00:17:59 --> 00:18:05
this special combination,
which is like the matrix A with

290
00:18:05 --> 00:18:10
lambda -- shifted by lambda,
shifted by lambda I,

291
00:18:10 --> 00:18:13
that it has to be singular.

292
00:18:13 --> 00:18:19
This matrix must be singular,
otherwise the only x would be

293
00:18:19 --> 00:18:23
the zero x, and zero matrix.OK.

294
00:18:23 --> 00:18:25
So this is singular.

295
00:18:25 --> 00:18:30
And what do I now know about
singular matrices?

296
00:18:30 --> 00:18:32
Their determinant is zero.

297
00:18:32 --> 00:18:38
So I've -- so from the fact
that that has to be singular,

298
00:18:38 --> 00:18:46
I know that the determinant of
A minus lambda I has to be zero.

299
00:18:46 --> 00:18:49
And that, now I've got x out of
it.

300
00:18:49 --> 00:18:54
I've got an equation for
lambda, that the key equation --

301
00:18:54 --> 00:19:00.35
it's called the characteristic
equation or the eigenvalue

302
00:19:00.35 --> 00:19:01
equation.

303
00:19:01 --> 00:19:06
And that -- in other words,
I'm now in a position to find

304
00:19:06 --> 00:19:09
lambda first.

305
00:19:09 --> 00:19:14
So -- the idea will be to find
lambda first.

306
00:19:14 --> 00:19:18.67
And actually,
I won't find one lambda,

307
00:19:18.67 --> 00:19:22
I'll find N different lambdas.

308
00:19:22 --> 00:19:27
Well, n lambdas,
maybe not n different

309
00:19:27 --> 00:19:28
ones.

310
00:19:28 --> 00:19:30
A lambda could be repeated.

311
00:19:30 --> 00:19:36
A repeated lambda is the source
of all trouble in 18.06.

312
00:19:36 --> 00:19:41
So, let's hope for the moment
that they're not repeated.

313
00:19:41 --> 00:19:45
There, there they were
different, right?

314
00:19:45 --> 00:19:51
One and minus one in that,
in that, for that permutation.

315
00:19:51 --> 00:19:51
OK.

316
00:19:51 --> 00:19:56
So and after I found this
lambda, can I just look ahead?

317
00:19:56 --> 00:19:59
How I going to find x?

318
00:19:59 --> 00:20:05
After I have found this lambda,
the lambda being this -- one of

319
00:20:05 --> 00:20:10
the numbers that makes this
matrix singular.

320
00:20:10 --> 00:20:13
Then of course finding x is
just by elimination.

321
00:20:13 --> 00:20:13
Right?

322
00:20:13 --> 00:20:17.03
It's just -- now I've got a
singular matrix,

323
00:20:17.03 --> 00:20:19
I'm looking for the null space.

324
00:20:19 --> 00:20:22
We're experts at finding the
null space.

325
00:20:22 --> 00:20:25
You know, you do elimination,
you identify the,

326
00:20:25 --> 00:20:28
the, the pivot columns and so
on,

327
00:20:28 --> 00:20:32
you're -- and,
give values to the free

328
00:20:32 --> 00:20:33
variables.

329
00:20:33 --> 00:20:37.83
Probably there'll only be one
free variable.

330
00:20:37.83 --> 00:20:41
We'll give it the value one,
like there.

331
00:20:41 --> 00:20:44
And we find the other variable.

332
00:20:44 --> 00:20:45
OK.

333
00:20:45 --> 00:20:51.21
So let's -- find the x second
will be a doable job.

334
00:20:51.21 --> 00:20:56.65
Let's go, let's look at the
first job of finding lambda.

335
00:20:56.65 --> 00:20:56
OK.

336
00:20:56 --> 00:20:59
Can I take another example?

337
00:20:59 --> 00:21:02
And let's, let's work that one
out.

338
00:21:02 --> 00:21:03
OK.

339
00:21:03 --> 00:21:08
So let me take the example,
say, let me make it easy.

340
00:21:08 --> 00:21:11
Three
three one and one.

341
00:21:11 --> 00:21:14
So I've made it easy.

342
00:21:14 --> 00:21:16
I've made it two by two.

343
00:21:16 --> 00:21:18
I've made it symmetric.

344
00:21:18 --> 00:21:23
And I even made it constant
down the diagonal.

345
00:21:23 --> 00:21:28
So that -- so the more,
like, special properties I

346
00:21:28 --> 00:21:34
stick into the matrix,
the more special outcome I

347
00:21:34 --> 00:21:36
get for the eigenvalues.

348
00:21:36 --> 00:21:40
For example,
this symmetric matrix,

349
00:21:40 --> 00:21:44
I know that it'll come out with
real eigenvalues.

350
00:21:44 --> 00:21:49
The eigenvalues will turn out
to be nice real numbers.

351
00:21:49 --> 00:21:56
And up in our previous example,
that was a symmetric matrix.

352
00:21:56 --> 00:22:00
Actually, while we're at it,
that was a symmetric matrix.

353
00:22:00 --> 00:22:05
Its eigenvalues were nice real
numbers, one and minus one.

354
00:22:05 --> 00:22:08
And do you notice anything
about its eigenvectors?

355
00:22:08 --> 00:22:13
Anything particular about those
two vectors, one one and minus

356
00:22:13 --> 00:22:15
one one?

357
00:22:15 --> 00:22:19
They just happen to be -- no,
I can't say they just happen to

358
00:22:19 --> 00:22:23
be, because that's the whole
point, is that they had to be --

359
00:22:23 --> 00:22:23.46
what?

360
00:22:23.46 --> 00:22:24
What are they?

361
00:22:24 --> 00:22:25
They're perpendicular.

362
00:22:25 --> 00:22:30.5
The vector, when I -- if I see
a vector one one and a one --

363
00:22:30.5 --> 00:22:34
and a minus one one,
my mind immediately takes that

364
00:22:34 --> 00:22:34
dot product.

365
00:22:34 --> 00:22:35
It's zero.

366
00:22:35 --> 00:22:37
Those vectors are
perpendicular.

367
00:22:37 --> 00:22:39
That'll happen here too.

368
00:22:39 --> 00:22:41
Well, let's find the
eigenvalues.

369
00:22:41 --> 00:22:45
Actually, oh,
my example's too easy.

370
00:22:45 --> 00:22:47
My example is too easy.

371
00:22:47 --> 00:22:51
Let me tell you in advance
what's going to happen.

372
00:22:51 --> 00:22:52.13
May I?

373
00:22:52.13 --> 00:22:56
Or shall I do the determinant
of A minus lambda,

374
00:22:56 --> 00:22:59
and then point out at the end?

375
00:22:59 --> 00:23:04
Will you remind me at the --
after I've found the

376
00:23:04 --> 00:23:11.05
eigenvalues to say why they
were -- why they were easy from

377
00:23:11.05 --> 00:23:14
the, from the example we did?

378
00:23:14 --> 00:23:17
OK, let's do the job here.

379
00:23:17 --> 00:23:22
Let's compute determinant of A
minus lambda I.

380
00:23:22 --> 00:23:25
So that's a determinant.

381
00:23:25 --> 00:23:29
And what's,
what is this thing?

382
00:23:29 --> 00:23:34
It's the matrix A with lambda
removed from the diagonal.

383
00:23:34 --> 00:23:40
So the diagonal matrix is
shifted, and then I'm taking the

384
00:23:40 --> 00:23:41
determinant.

385
00:23:41 --> 00:23:42
OK.

386
00:23:42 --> 00:23:44.45
So I multiply this out.

387
00:23:44.45 --> 00:23:47
So what is that determinant?

388
00:23:47 --> 00:23:52
Do you notice,
I didn't take lambda away from

389
00:23:52 --> 00:23:54
all the entries.

390
00:23:54 --> 00:23:57
It's lambda I,
so it's lambda along the

391
00:23:57 --> 00:23:58
diagonal.

392
00:23:58 --> 00:24:03
So I get three minus lambda
squared and then minus one,

393
00:24:03 --> 00:24:04
right?

394
00:24:04 --> 00:24:08
And I want that to be zero.

395
00:24:08 --> 00:24:13
Well, I'm going to simplify it.

396
00:24:13 --> 00:24:16
And what will I get?

397
00:24:16 --> 00:24:25.3
So if I multiply this out,
I get lambda squared minus six

398
00:24:25.3 --> 00:24:28
lambda plus what?

399
00:24:28 --> 00:24:29
Plus eight.

400
00:24:29 --> 00:24:35
And that I'm going to set to
zero.

401
00:24:35 --> 00:24:39
And I'm
going to solve it.

402
00:24:39 --> 00:24:42
So and it's,
it's a quadratic equation.

403
00:24:42 --> 00:24:45
I can use factorization,
I can use the quadratic

404
00:24:45 --> 00:24:46
formula.

405
00:24:46 --> 00:24:48
I'll get two lambdas.

406
00:24:48 --> 00:24:51
Before I do it,
tell me what's that number six

407
00:24:51 --> 00:24:55
that's showing up in this
equation?

408
00:24:55 --> 00:24:56
It's the trace.

409
00:24:56 --> 00:24:59
That number six is three plus
three.

410
00:24:59 --> 00:25:03
And while we're at it,
what's the number eight that's

411
00:25:03 --> 00:25:06
showing up in this equation?

412
00:25:06 --> 00:25:08
It's the determinant.

413
00:25:08 --> 00:25:11.41
That our matrix has determinant
eight.

414
00:25:11.41 --> 00:25:15.95
So in a two by two case,
it's really nice.

415
00:25:15.95 --> 00:25:20
It's lambda squared minus the
trace times lambda -- the trace

416
00:25:20 --> 00:25:25
is the linear coefficient -- and
plus the determinant,

417
00:25:25 --> 00:25:26
the constant term.

418
00:25:26 --> 00:25:26
OK.

419
00:25:26 --> 00:25:29
So let's -- can,
can we find the roots?

420
00:25:29 --> 00:25:34
I guess the easy way is to
factor that as something times

421
00:25:34 --> 00:25:36
something.

422
00:25:36 --> 00:25:41
If we couldn't factor it,
then we'd have to use the old

423
00:25:41 --> 00:25:45
b^2-4ac formula,
but I, I think we can factor

424
00:25:45 --> 00:25:50
that into lambda minus what
times lambda minus what?

425
00:25:50 --> 00:25:53
Can you do that factorization?

426
00:25:53 --> 00:25:54
Four and two?

427
00:25:54 --> 00:25:59
Lambda minus four times lambda
minus two.

428
00:25:59 --> 00:26:02.61
So the, the eigenvalues are
four and two.

429
00:26:02.61 --> 00:26:07
So the eigenvalues are -- one
eigenvalue, lambda one,

430
00:26:07 --> 00:26:09
let's say, is four.

431
00:26:09 --> 00:26:12
Lambda two, the other
eigenvalue, is two.

432
00:26:12 --> 00:26:15
The eigenvalues are four and
two.

433
00:26:15 --> 00:26:19
And then I can go for the
eigenvectors.

434
00:26:19 --> 00:26:23
You see I got the eigenvalues
first.

435
00:26:23 --> 00:26:24
Four and two.

436
00:26:24 --> 00:26:26
Now for the eigenvectors.

437
00:26:26 --> 00:26:29
So what are the eigenvectors?

438
00:26:29 --> 00:26:34
They're these guys in the null
space when I take away,

439
00:26:34 --> 00:26:38
when I make the matrix singular
by

440
00:26:38 --> 00:26:41
taking four I or two I away.

441
00:26:41 --> 00:26:45
So we're -- we got to do those
separately.

442
00:26:45 --> 00:26:51
I'll -- let me find the
eigenvector for four first.

443
00:26:51 --> 00:26:56
So I'll subtract four,
so A minus four I is -- so

444
00:26:56 --> 00:27:00
taking four away will put minus
ones there.

445
00:27:00 --> 00:27:05
And what's the point
about that matrix?

446
00:27:05 --> 00:27:09.84
If four is an eigenvalue,
then A minus four I had better

447
00:27:09.84 --> 00:27:11
be a what kind of matrix?

448
00:27:11 --> 00:27:12
Singular.

449
00:27:12 --> 00:27:16
If that matrix isn't singular,
the four wasn't correct.

450
00:27:16 --> 00:27:19
But we're OK,
that matrix is singular.

451
00:27:19 --> 00:27:21
And what's the x now?

452
00:27:21 --> 00:27:24
The x is in the null
space.

453
00:27:24 --> 00:27:30.08
So what's the x1 that goes
with, with the lambda one?

454
00:27:30.08 --> 00:27:36
So that A -- so this is -- now
I'm doing A x1 is lambda one x1.

455
00:27:36 --> 00:27:42
So I took A minus lambda one I,
that's this matrix,

456
00:27:42 --> 00:27:47
and now I'm looking for the x1
in its null space,

457
00:27:47 --> 00:27:48
and who is he?

458
00:27:48 --> 00:27:51
What's the vector x in the null
space?

459
00:27:51 --> 00:27:53
Of course it's one one.

460
00:27:53 --> 00:27:57
So that's the eigenvector that
goes with that eigenvalue.

461
00:27:57 --> 00:28:01
Now how about the eigenvector
that goes with the other

462
00:28:01 --> 00:28:02.7
eigenvalue?

463
00:28:02.7 --> 00:28:05
Can I do that with,
with erasing?

464
00:28:05 --> 00:28:07
I take A minus two I.

465
00:28:07 --> 00:28:11.94
So now I take two away from the
diagonal, and that leaves me

466
00:28:11.94 --> 00:28:13
with a one and a one.

467
00:28:13 --> 00:28:18
So A minus two I has again
produced a singular matrix,

468
00:28:18 --> 00:28:19
as it had to.

469
00:28:19 --> 00:28:23
I'm looking for the null space
of that guy.

470
00:28:23 --> 00:28:25
What vector is in its null
space?

471
00:28:25 --> 00:28:28
Well, of course,
a whole line of vectors.

472
00:28:28 --> 00:28:32
So when I say the eigenvector,
I'm not speaking correctly.

473
00:28:32 --> 00:28:36
There's a whole line of
eigenvectors,

474
00:28:36 --> 00:28:39
and you just -- I just want a
basis.

475
00:28:39 --> 00:28:41
And for a line I just want one
vector.

476
00:28:41 --> 00:28:45
But -- You could,
you're -- there's some freedom

477
00:28:45 --> 00:28:49.14
in choosing that one,
but choose a reasonable one.

478
00:28:49.14 --> 00:28:53
What's a vector in the null
space of that?

479
00:28:53 --> 00:28:58
Well, the natural vector to
pick as the eigenvector with,

480
00:28:58 --> 00:29:00
with lambda two is minus one
one.

481
00:29:00 --> 00:29:05
If I did elimination on that
vector and set that,

482
00:29:05 --> 00:29:10.53
the free variable to be one,
I would get minus one and get

483
00:29:10.53 --> 00:29:12
that eigenvector.

484
00:29:12 --> 00:27:24
So you see then that
I've got eigenvector,

485
00:27:24 --> 00:23:32
eigenvalue, eigenvector,
eigenvalue for this,

486
00:23:32 --> 00:22:08
for this matrix?

487
00:22:08 --> 00:17:13
And now comes that thing that I
wanted to be reminded of.

488
00:17:13 --> 00:12:02.24
What's the relation between
that problem and -- let me write

489
00:12:02.24 --> 00:10:44
just above what we2
found here.

490
00:10:44 --> 00:15:59
A equals zero one one zero,
that had eigenvalue one and

491
00:15:59 --> 00:21:43
minus one and eigenvectors one
one and eigenvector minus one

492
00:21:43 --> 00:22:06
one.

493
00:22:06 --> 00:24:20
And what do you notice?

494
00:24:20 --> 00:30:11
What's -- how is this matrix
related to that matrix?

495
00:30:11 --> 00:30:14
How are those two matrices
related?

496
00:30:14 --> 00:30:18
Well, one is just three I more
than the other one,

497
00:30:18 --> 00:30:19
right?

498
00:30:19 --> 00:30:24
I just took that matrix and I
-- I took this matrix and I

499
00:30:24 --> 00:30:25
added three I.

500
00:30:25 --> 00:30:28
So my question is,
what happened to the

501
00:30:28 --> 00:30:32
eigenvalues and
what happened to the

502
00:30:32 --> 00:30:33
eigenvectors?

503
00:30:33 --> 00:30:36
That's the, that's like the
question we keep asking now in

504
00:30:36 --> 00:30:37
this chapter.

505
00:30:37 --> 00:30:41
If I, if I do something to the
matrix, what happens if I -- or

506
00:30:41 --> 00:30:45
I know something about the
matrix, what's the what's the

507
00:30:45 --> 00:30:48
conclusion for its eigenvectors
and

508
00:30:48 --> 00:30:49
eigenvalues?

509
00:30:49 --> 00:30:53
Because -- those eigenvalues
and eigenvectors are going to

510
00:30:53 --> 00:30:57.28
tell us important information
about the matrix.

511
00:30:57.28 --> 00:30:59
And here what are we seeing?

512
00:30:59 --> 00:31:03
What's happening to these
eigenvalues, one and minus one,

513
00:31:03 --> 00:31:06
when I add three
I?

514
00:31:06 --> 00:31:09
It just added three to the
eigenvalues.

515
00:31:09 --> 00:31:13
I got four and two,
three more than one and minus

516
00:31:13 --> 00:31:13
one.

517
00:31:13 --> 00:31:16
What happened to the
eigenvectors?

518
00:31:16 --> 00:31:17
Nothing at all.

519
00:31:17 --> 00:31:22.75
One one is -- and minus -- and
one -- and minus

520
00:31:22.75 --> 00:31:26
one one are -- is still the
eigenvectors.

521
00:31:26 --> 00:31:30
In other words,
simple but useful observation.

522
00:31:30 --> 00:31:36
If I add three I to a matrix,
its eigenvectors don't change

523
00:31:36 --> 00:31:39.92
and its eigenvalues are three
bigger.

524
00:31:39.92 --> 00:31:42
Let's, let's just see why.

525
00:31:42 --> 00:26:15
Let me keep all
this on the same board.

526
00:26:15 --> 00:14:48
Suppose I have a matrix A,
and Ax equal lambda x.

527
00:14:48 --> 00:07:10
Now I add three I to that
matrix.

528
00:07:10 --> 00:05:46
Do you see what3
so it's if Ax equals lambda x,

529
00:05:46 --> 00:11:54
then this, this other new
matrix, I just have an Ax,

530
00:11:54 --> 00:16:36
which is lambda x,
and I have a three x,

531
00:16:36 --> 00:21:03
from the three x,
so it's just I mean,

532
00:21:03 --> 00:23:56
it's just sitting there.

533
00:23:56 --> 00:26:20
Lambda plus three x.

534
00:26:20 --> 00:32:36
So if they, if this had
eigenvalue lambda,

535
00:32:36 --> 00:32:40
this has eigenvalue lambda plus
three.

536
00:32:40 --> 00:32:45
And x, the eigenvector,
is the same x for both

537
00:32:45 --> 00:32:46.84
matrices.

538
00:32:46.84 --> 00:32:47
OK.

539
00:32:47 --> 00:32:49
So that's, great.

540
00:32:49 --> 00:32:52
Of course, it's special.

541
00:32:52 --> 00:32:57.75
We got the new matrix by adding
three I.

542
00:32:57.75 --> 00:33:01
Suppose I had added another
matrix.

543
00:33:01 --> 00:33:07
Suppose I know the eigenvalues
and eigenvectors of A.

544
00:33:07 --> 00:33:13
So this is, this,
this little board here is going

545
00:33:13 --> 00:33:15
to be not so great.

546
00:33:15 --> 00:33:23
Suppose I have a matrix A and
it has an eigenvector x with

547
00:33:23 --> 00:33:25
an eigenvalue lambda.

548
00:33:25 --> 00:33:28
And now I add on some other
matrix.

549
00:33:28 --> 00:33:33
So, so what I'm asking you is,
if you know the eigenvalues of

550
00:33:33 --> 00:33:39.59
A and you know the eigenvalues
of B, let me say suppose B -- so

551
00:33:39.59 --> 00:33:42
this is if -- let me put an if
here.

552
00:33:42 --> 00:33:46
If Ax equals lambda x,
fine, and B has,

553
00:33:46 --> 00:33:52
eigenvalues,
has eigenvalues -- what shall

554
00:33:52 --> 00:33:54
we call them?

555
00:33:54 --> 00:34:02
Alpha, alpha one and alpha --
let's say -- I'll use alpha for

556
00:34:02 --> 00:34:08
the eigenvalues of B for no good
reason.

557
00:34:08 --> 00:34:18
What a- you see what I'm going
to ask is, how about A plus B?

558
00:34:18 --> 00:34:22
Let me, let me give you the,
let me give you,

559
00:34:22 --> 00:34:25
what you might think first.

560
00:34:25 --> 00:34:26
OK.

561
00:34:26 --> 00:34:32.31
If Ax equals lambda x and if B
has an eigenvalue alpha,

562
00:34:32.31 --> 00:34:39.13
then I allowed to say -- what's
the matter with this argument?

563
00:34:39.13 --> 00:34:41
It's wrong.

564
00:34:41 --> 00:34:44.42
What I'm going to write up is
wrong.

565
00:34:44.42 --> 00:34:47
I'm going to say Bx is alpha x.

566
00:34:47 --> 00:34:50
Add those up,
and you get A plus B x equals

567
00:34:50 --> 00:34:52
lambda plus alpha x.

568
00:34:52 --> 00:34:57
So you would think that if you
know the eigenvalues of A and

569
00:34:57 --> 00:35:02.66
you knew the eigenvalues of B,
then if you added you would

570
00:35:02.66 --> 00:35:06.5
know the eigenvalues of A plus
B.

571
00:35:06.5 --> 00:35:08
But that's false.

572
00:35:08 --> 00:35:13
A plus B -- well,
when B was three I,

573
00:35:13 --> 00:35:15
that worked great.

574
00:35:15 --> 00:35:19
But this is not so great.

575
00:35:19 --> 00:35:25
And what's the matter with that
argument there?

576
00:35:25 --> 00:35:33.05
We have no reason to believe
that x is also an eigenvector of

577
00:35:33.05 --> 00:35:33
B.

578
00:35:33 --> 00:35:36.94
B has some
eigenvalues,

579
00:35:36.94 --> 00:35:40.48
but it's got some different
eigenvectors normally.

580
00:35:40.48 --> 00:35:42
It's a different matrix.

581
00:35:42 --> 00:35:44
I don't know anything special.

582
00:35:44 --> 00:35:48
If I don't know anything
special, then as far as I know,

583
00:35:48 --> 00:35:51
it's got some different
eigenvector y,

584
00:35:51 --> 00:35:54
and when I add I
get just rubbish.

585
00:35:54 --> 00:36:00
I mean, I get -- I can add,
but I don't learn anything.

586
00:36:00 --> 00:36:03
So not so great is A plus B.

587
00:36:03 --> 00:36:05
Or A times B.

588
00:36:05 --> 00:36:11
Normally the eigenvalues of A
plus B or A times B are not

589
00:36:11 --> 00:36:16.32
eigenvalues of A plus
eigenvalues of B.

590
00:36:16.32 --> 00:36:19
Ei- eigenvalues are not,
like, linear.

591
00:36:19 --> 00:36:22
Or -- and they don't multiply.

592
00:36:22 --> 00:36:27
Because, eigenvectors are
usually different and,

593
00:36:27 --> 00:36:32
and there's just no way to find
out what A plus B does to affect

594
00:36:32 --> 00:36:33
it.

595
00:36:33 --> 00:36:33
OK.

596
00:36:33 --> 00:36:37
So that's, like,
a caution.

597
00:36:37 --> 00:36:40
Don't, if B is a multiple of
the identity,

598
00:36:40 --> 00:36:47
great, but if B is some general
matrix, then for A plus B you've

599
00:36:47 --> 00:36:52.51
got to find -- you've got to
solve the eigenvalue problem.

600
00:36:52.51 --> 00:36:52
OK.

601
00:36:52 --> 00:36:58
Now I want to do another
example that brings out a,

602
00:36:58 --> 00:37:02
another point about
eigenvalues.

603
00:37:02 --> 00:37:08.44
Let me make this example a
rotation matrix.

604
00:37:08.44 --> 00:37:08.85
OK.

605
00:37:08.85 --> 00:37:12
So here's another example.

606
00:37:12 --> 00:37:17
So a rotate -- oh,
I'd better call it Q.

607
00:37:17 --> 00:37:23.99
I often use Q for,
for, rotations because those

608
00:37:23.99 --> 00:37:29
are the, like,
very important examples of

609
00:37:29 --> 00:37:31
orthogonal matrices.

610
00:37:31 --> 00:37:35
Let me make it a ninety degree
rotation.

611
00:37:35 --> 00:37:41
So -- my matrix is going to be
the one that rotates every

612
00:37:41 --> 00:37:43
vector by ninety degrees.

613
00:37:43 --> 00:37:47
So do you remember
that matrix?

614
00:37:47 --> 00:37:52
It's the cosine of ninety
degrees, which is zero,

615
00:37:52 --> 00:37:57
the sine of ninety degrees,
which is one,

616
00:37:57 --> 00:38:02
minus the sine of ninety,
the cosine of ninety.

617
00:38:02 --> 00:38:06
So that matrix deserves the
letter Q.

618
00:38:06 --> 00:38:12
It's an orthogonal matrix,
very, very orthogonal matrix.

619
00:38:12 --> 00:38:17
Now I'm interested in its
eigenvalues and eigenvectors.

620
00:38:17 --> 00:38:21
Two by two, it can't be that
tough.

621
00:38:21 --> 00:38:24
We know that the eigenvalues
add to zero.

622
00:38:24 --> 00:38:28
Actually, we know something
already here.

623
00:38:28 --> 00:38:33
The eigen- what's the sum of
the two eigenvalues?

624
00:38:33 --> 00:38:36
Just tell
me what I just said.

625
00:38:36 --> 00:38:37
Zero, right.

626
00:38:37 --> 00:38:39
From that trace business.

627
00:38:39 --> 00:38:43
The sum of the eigenvalues is,
is going to come out zero.

628
00:38:43 --> 00:38:47
And the product of the
eigenvalues, did I tell you

629
00:38:47 --> 00:38:51.39
about the determinant
being the product of the

630
00:38:51.39 --> 00:38:52
eigenvalues?

631
00:38:52 --> 00:38:52
No.

632
00:38:52 --> 00:38:55
But that's a good thing to
know.

633
00:38:55 --> 00:38:59
We pointed out how that eight
appeared in the,

634
00:38:59 --> 00:39:02
in the quadratic equation.

635
00:39:02 --> 00:39:04
So let me just say this.

636
00:39:04 --> 00:37:36
The trace is zero plus zero,
obviously.

637
00:37:36 --> 00:30:18
And that's the sum,
that's lambda one plus lambda

638
00:30:18 --> 00:29:42
two.

639
00:29:42 --> 00:22:34
Now the other neat fact is that
the determinant,

640
00:22:34 --> 00:16:56.71
what's the determinant of that
matrix?

641
00:16:56.71 --> 00:16:20
One.

642
00:16:20 --> 00:09:30
And that is lambda one times
lambda3

643
00:09:30 --> 00:10:00
two.

644
00:10:00 --> 00:14:40
In our example,
the one we worked out,

645
00:14:40 --> 00:20:06.03
we -- the eigenvalues came out
four and two.

646
00:20:06.03 --> 00:23:07
Their product was eight.

647
00:23:07 --> 00:30:26
That -- it had to be eight,
because we factored into lambda

648
00:30:26 --> 00:34:36
minus four times lambda minus
two.

649
00:34:36 --> 00:39:48
That gave us the constant term
eight.

650
00:39:48 --> 00:39:51
And that was the determinant.

651
00:39:51 --> 00:39:52
OK.

652
00:39:52 --> 00:40:00
What I'm leading up to with
this example is that something's

653
00:40:00 --> 00:40:02
going to go wrong.

654
00:40:02 --> 00:40:09
Something goes wrong for
rotation because what vector can

655
00:40:09 --> 00:40:17
come out parallel to itself
after a rotation?

656
00:40:17 --> 00:40:22.23
If this matrix rotates every
vector by ninety degrees,

657
00:40:22.23 --> 00:40:25
what could be an eigenvector?

658
00:40:25 --> 00:40:29
Do you see we're,
we're, we're going to have

659
00:40:29 --> 00:40:32
trouble.
eigenvectors are -- Well.

660
00:40:32 --> 00:40:35.75
Our, our picture of
eigenvectors,

661
00:40:35.75 --> 00:40:41
of, of coming out in the same
direction that they went in,

662
00:40:41 --> 00:40:43.41
there won't be it.

663
00:40:43.41 --> 00:40:48
And with, and with eigenvalues
we're going to have trouble.

664
00:40:48 --> 00:40:49
From these equations.

665
00:40:49 --> 00:40:50
Let's see.

666
00:40:50 --> 00:40:52
Why I expecting trouble?

667
00:40:52 --> 00:40:56.5
The, the first equation says
that the

668
00:40:56.5 --> 00:40:58
eigenvalues add to zero.

669
00:40:58 --> 00:41:01.91
So there's a plus and a minus.

670
00:41:01.91 --> 00:41:07
But then the second equation
says that the product is plus

671
00:41:07 --> 00:41:08
one.

672
00:41:08 --> 00:41:09
We're in trouble.

673
00:41:09 --> 00:41:11
But there's a way out.

674
00:41:11 --> 00:41:15
So how -- let's do the usual
stuff.

675
00:41:15 --> 00:41:20
Look at determinant of Q minus
lambda I.

676
00:41:20 --> 00:41:25
So I'll just follow the rules,
take the determinant,

677
00:41:25 --> 00:41:29
subtract lambda from the
diagonal, where I had zeros,

678
00:41:29 --> 00:41:31
the rest is the same.

679
00:41:31 --> 00:41:34
Rest of Q is just copied.

680
00:41:34 --> 00:41:36
Compute that determinant.

681
00:41:36 --> 00:41:40
OK, so what does that
determinant equal?

682
00:41:40 --> 00:41:45
Lambda squared minus minus one
plus what?

683
00:41:45 --> 00:41:46
What's up?

684
00:41:46 --> 00:41:49
There's my equation.

685
00:41:49 --> 00:41:56
My equation for the eigenvalues
is lambda squared plus one

686
00:41:56 --> 00:41:57
equals zero.

687
00:41:57 --> 00:42:03
What are the eigenvalues lambda
one

688
00:42:03 --> 00:42:07
and lambda two?

689
00:42:07 --> 00:42:19
They're I, whatever that is,
and minus it,

690
00:42:19 --> 00:42:21
right.

691
00:42:21 --> 00:42:30.75
Those are the right numbers.

692
00:42:30.75 --> 00:42:36
To be real numbers even though
the matrix was perfectly real.

693
00:42:36 --> 00:42:38
So this can happen.

694
00:42:38 --> 00:42:45
Complex numbers are going to --
have to enter eighteen oh six at

695
00:42:45 --> 00:42:46
this moment.

696
00:42:46 --> 00:42:47
Boo, right.

697
00:42:47 --> 00:42:48
All right.

698
00:42:48 --> 00:39:01.15
If I just choose good
matrices that have real

699
00:39:01.15 --> 00:31:59
eigenvalues, we can postpone
that evil day,

700
00:31:59 --> 00:24:28
but just so you see -- so I'll
try to do that.

701
00:24:28 --> 00:21:18
But it's out there.

702
00:21:18 --> 00:14:06
That a matrix,
a perfectly real matrix could

703
00:14:06 --> 00:06:05
have, give a perfectly
innocent-looking quadratic

704
00:06:05 --> 00:08:14.67
thing,4
but the roots of that quadratic

705
00:08:14.67 --> 00:11:57
can be complex numbers.

706
00:11:57 --> 00:21:08
And of course you -- everybody
knows that they're -- what,

707
00:21:08 --> 00:27:54.02
what do you know about the
complex numbers?

708
00:27:54.02 --> 00:37:05
So, so now -- Let's just spend
one more minute on this bad

709
00:37:05 --> 00:43:31
possibility of complex numbers.

710
00:43:31 --> 00:43:37
We do know a little information
about the, the two complex

711
00:43:37 --> 00:43:38.08
numbers.

712
00:43:38.08 --> 00:43:42
They're complex conjugates of
each other.

713
00:43:42 --> 00:43:46
If, if lambda is an eigenvalue,
then when I change,

714
00:43:46 --> 00:43:53
when I go -- you remember what
complex conjugates are?

715
00:43:53 --> 00:43:56
You switch the sign of the
imaginary part.

716
00:43:56 --> 00:43:59
Well, this was only imaginary,
had no real part,

717
00:43:59 --> 00:44:02
so we just switched its sign.

718
00:44:02 --> 00:44:05
So that eigenvalues come in
pairs like that,

719
00:44:05 --> 00:44:06
but they're complex.

720
00:44:06 --> 00:44:08
A complex conjugate pair.

721
00:44:08 --> 00:44:13
And that can happen
with a perfectly real matrix.

722
00:44:13 --> 00:44:17
And as a matter of fact -- so
that was my, my point earlier,

723
00:44:17 --> 00:44:21
that if a matrix was symmetric,
it wouldn't happen.

724
00:44:21 --> 00:44:25.57
So if we stick to matrices that
are symmetric or,

725
00:44:25.57 --> 00:44:29
like, close to symmetric,
then the eigenvalues will stay

726
00:44:29 --> 00:44:30
real.

727
00:44:30 --> 00:44:35
But if we move far away
from symmetric -- and that's as

728
00:44:35 --> 00:44:39
far as you can move,
because that matrix is -- how

729
00:44:39 --> 00:44:42
is Q transpose related to Q for
that matrix?

730
00:44:42 --> 00:44:45
That matrix is anti-symmetric.

731
00:44:45 --> 00:44:46
Q transpose is minus Q.

732
00:44:46 --> 00:44:50.5
That's the very opposite of
symmetry.

733
00:44:50.5 --> 00:44:54
When I flip across the diagonal
I get -- I reverse all the

734
00:44:54 --> 00:44:55
signs.

735
00:44:55 --> 00:44:59
Those are the guys that have
pure imaginary eigenvalues.

736
00:44:59 --> 00:45:01
So they're the extreme case.

737
00:45:01 --> 00:45:05
And in between are,
are matrices that are not

738
00:45:05 --> 00:45:09
symmetric or anti-symmetric but,
but they have partly a

739
00:45:09 --> 00:45:13
symmetric
part and an anti-symmetric

740
00:45:13 --> 00:45:13
part.

741
00:45:13 --> 00:45:14.2
OK.

742
00:45:14.2 --> 00:45:19
So I'm doing a bunch of
examples here to show the

743
00:45:19 --> 00:45:20
possibilities.

744
00:45:20 --> 00:45:26
The good possibilities being
perpendicular eigenvectors,

745
00:45:26 --> 00:45:28
real eigenvalues.

746
00:45:28 --> 00:45:34
The bad possibilities being
complex eigenvalues.

747
00:45:34 --> 00:45:37
We could say that's bad.

748
00:45:37 --> 00:45:40
There's another even worse.

749
00:45:40 --> 00:45:46
I'm getting through the,
the bad things here today.

750
00:45:46 --> 00:45:53
Then, then the next lecture
can, can, can be like pure

751
00:45:53 --> 00:45:54
happiness.

752
00:45:54 --> 00:45:54
OK.

753
00:45:54 --> 00:46:00
Here's one more bad thing
that could happen.

754
00:46:00 --> 00:46:04
So I, again,
I'll do it with an example.

755
00:46:04 --> 00:46:09
Suppose my matrix is,
suppose I take this three three

756
00:46:09 --> 00:46:13
one and I change that guy to
zero.

757
00:46:13 --> 00:46:17.18
What are the eigenvalues of
that matrix?

758
00:46:17.18 --> 00:46:19
What are the eigenvectors?

759
00:46:19 --> 00:46:23
This is
always our question.

760
00:46:23 --> 00:46:27.54
Of course, the next section
we're going to show why are,

761
00:46:27.54 --> 00:46:28
why do we care.

762
00:46:28 --> 00:46:32
But for the moment,
this lecture is introducing

763
00:46:32 --> 00:46:32
them.

764
00:46:32 --> 00:46:34
And let's just find them.

765
00:46:34 --> 00:46:34
OK.

766
00:46:34 --> 00:46:38
What are the eigenvalues of
that matrix?

767
00:46:38 --> 00:46:43
Let me tell you -- at a glance
we could answer that question.

768
00:46:43 --> 00:46:46
Because the matrix is
triangular.

769
00:46:46 --> 00:46:51
It's really useful to know --
if you've got properties like a

770
00:46:51 --> 00:46:52
triangular matrix.

771
00:46:52 --> 00:46:57
It's very useful to know you
can read the eigenvalues

772
00:46:57 --> 00:46:57
off.

773
00:46:57 --> 00:47:00
They're right on the diagonal.

774
00:47:00 --> 00:47:03
So the eigenvalue is three and
also three.

775
00:47:03 --> 00:47:05.43
Three is a repeated eigenvalue.

776
00:47:05.43 --> 00:47:07
But let's see that happen.

777
00:47:07 --> 00:47:08
Let me do it right.

778
00:47:08 --> 00:47:12
The determinant of A minus
lambda I, what I always have to

779
00:47:12 --> 00:47:15
do is this
determinant.

780
00:47:15 --> 00:47:19
I take away lambda from the
diagonal.

781
00:47:19 --> 00:47:21
I leave the rest.

782
00:47:21 --> 00:47:27
I compute the determinant,
so I get a three minus lambda

783
00:47:27 --> 00:47:30
times a three minus lambda.

784
00:47:30 --> 00:47:31
And nothing.

785
00:47:31 --> 00:47:37
So that's where the triangular
part came in.

786
00:47:37 --> 00:47:40
Triangular part,
the one thing we know about

787
00:47:40 --> 00:47:44
triangular matrices is the
determinant is just the product

788
00:47:44 --> 00:47:46
down the diagonal.

789
00:47:46 --> 00:47:48
And in this case,
it's this same,

790
00:47:48 --> 00:47:51
repeated -- so lambda one is
one -- sorry,

791
00:47:51 --> 00:47:56
lambda one is three and lambda
two is three.

792
00:47:56 --> 00:47:57
That was easy.

793
00:47:57 --> 00:48:02
I mean, no -- why should I be
pessimistic about a matrix whose

794
00:48:02 --> 00:48:06.07
eigenvalues can be read off
right away?

795
00:48:06.07 --> 00:48:10
The problem with this matrix is
in the eigenvectors.

796
00:48:10 --> 00:48:13
So let's go to the
eigenvectors.

797
00:48:13 --> 00:48:17.3
So how do I find the
eigenvectors?

798
00:48:17.3 --> 00:41:01
I'm looking for a couple of
eigenvectors.

799
00:41:01 --> 00:36:28
So I take the eigenvalue.

800
00:36:28 --> 00:33:23
What do I do now?

801
00:33:23 --> 00:26:17
You remember,
I solve A minus lambda I x

802
00:26:17 --> 00:24:06
equals zero.

803
00:24:06 --> 00:18:28
And what is A minus lambda I x?

804
00:18:28 --> 00:14:18
So, so take three away.

805
00:14:18 --> 00:12:07
And I get this matrix4
zero zero zero one,

806
00:12:07 --> 00:13:12
right?

807
00:13:12 --> 00:20:50
Times x is supposed to give me
zero, right?

808
00:20:50 --> 00:26:06
That's my big equation for x.

809
00:26:06 --> 00:33:01.01
Now I'm looking for x,
the eigenvector.

810
00:33:01.01 --> 00:43:11
So I took A minus lambda I x,
and what kind of a matrix I

811
00:43:11 --> 00:49:00.48
supposed to have here?

812
00:49:00.48 --> 00:49:01
Singular, right?

813
00:49:01 --> 00:49:03
It's supposed to be singular.

814
00:49:03 --> 00:49:07
And then it's got some vectors
-- which it is.

815
00:49:07 --> 00:49:10
So it's got some vector x in
the null space.

816
00:49:10 --> 00:49:14
And what, what's the,
what's -- give me a basis for

817
00:49:14 --> 00:49:17
the null space for that guy.

818
00:49:17 --> 00:49:21
Tell me, what's a vector x in
the null space,

819
00:49:21 --> 00:49:26
so that'll be the,
the eigenvector that goes with

820
00:49:26 --> 00:49:28
lambda one equals three.

821
00:49:28 --> 00:49:33
The eigenvector is -- so what's
in the null space?

822
00:49:33 --> 00:49:35
One zero, is it?

823
00:49:35 --> 00:49:37
Great.

824
00:49:37 --> 00:49:39
Now, what's the other
eigenvector?

825
00:49:39 --> 00:49:44
What's, what's the eigenvector
that goes with lambda two?

826
00:49:44 --> 00:49:47
Well, lambda two is three
again.

827
00:49:47 --> 00:49:50
So I get the same thing again.

828
00:49:50 --> 00:49:54
Give me another vector -- I
want it to be independent.

829
00:49:54 --> 00:50:00.44
If I'm going to write
down an x2, I'm never going to

830
00:50:00.44 --> 00:50:03
let it be dependent on x1.

831
00:50:03 --> 00:50:07
I'm looking for independent
eigenvectors,

832
00:50:07 --> 00:50:10
and what's the conclusion?

833
00:50:10 --> 00:50:11
There isn't one.

834
00:50:11 --> 00:50:14.66
This is a degenerate matrix.

835
00:50:14.66 --> 00:50:21
It's only got one line of
eigenvectors instead of two.

836
00:50:21 --> 00:50:28
It's this possibility of a
repeated eigenvalue opens this

837
00:50:28 --> 00:50:34
further possibility of a
shortage of eigenvectors.

838
00:50:34 --> 00:50:40
And so there's no second
independent eigenvector x2.

839
00:50:40 --> 00:50:46.09
So it's a matrix,
it's a two by two matrix,

840
00:50:46.09 --> 00:50:49
but with
only one independent

841
00:50:49 --> 00:50:50
eigenvector.

842
00:50:50 --> 00:50:56
So that will be -- the matrices
that -- where eigenvectors are

843
00:50:56 --> 00:50:58
-- don't give the complete
story.

844
00:50:58 --> 00:50:59
OK.

845
00:50:59 --> 00:51:04
My lecture on Monday will give
the complete story for all the

846
00:51:04 --> 00:51:06
other matrices.

847
00:51:06 --> 00:51:07
Thanks.

848
00:51:07 --> 00:51:08.96
Have a good weekend.

849
00:51:08.96 --> 00:51:11
A real New England weekend.