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PROFESSOR STRANG:
Shall we start?

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The main job of today is
eigenvalues and eigenvectors.

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The next section in the book
and a very big topic and

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things to say about it.
 

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00:00:37 --> 00:00:40
I do want to begin with a
recap of what I didn't

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quite finish last time.
 

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So what we did was solve this
very straightforward equation.

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Straightforward except that
it has a point source,

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a delta function.
 

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And we solved it, both the
fixed-fixed case when a

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straight line went up and back
down and in the free-fixed case

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when it was a horizontal line
and then down with

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slope minus one.
 

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And there are different ways
to get to this answer.

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But once you have it,
you can look at it and

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say, well is it right?
 

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Certainly the boundary
conditions are correct.

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Zero slope went through
zero, that's good.

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And then the only thing you
really have to check is does

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the slope drop by one at the
point of the impulse? because

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that's what this is
forcing us to do.

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It's saying the slope
should drop by one.

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And here the slope
is 1-a going up.

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And if I take the derivative,
it's -a going down.

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1-a dropped to -a, good.
 

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Here the slope was zero.
 

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Here the slope was
minus one, good.

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So those are the right answers.
 

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00:02:01 --> 00:02:11
And this is simple, but
really a great example.

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And then, what I wanted
to do was catch the same

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thing for the matrices.
 

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So those matrices, we all know
what K is and what T is.

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So I'm solving, I'm
really solving K

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inverse equal identity.
 

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That's the equation
I'm solving.

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So I'm looking for K inverse
and trying to get the

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columns of the identity.
 

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00:02:40 --> 00:02:44
And you realize the columns
of the identity are just

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like delta vectors.
 

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They've got a one in one
spot, they're a point load

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just like this thing.
 

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So can I just say how
I remember K inverse?

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I finally, you know, again
there are different

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ways to get to it.
 

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One way is MATLAB, just do it.
 

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00:03:03 --> 00:03:10
But I guess maybe the whole
point is, the whole point of

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these and the eigenvalues that
are coming too, is this.

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That we have here the chance
to see important, special

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cases that work out.
 

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Normally we don't find the
inverse, print out the

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inverse of a matrix.
 

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It's not nice.
 

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Normally we just let eig
find the eigenvalues.

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Because that's an even worse
calculation, to find

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eigenvalues, in general.
 

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I'm talking here about our
matrices of all sizes n by n.

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Nobody finds the eigenvalues
by hand of n by n matrices.

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But these have terrific
eigenvalues and

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especially eigenvectors.
 

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So in a way this is a little
bit like, typical of math.

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That you ask about general
stuff or you write the

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equation with a matrix A.
 

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So that's the general
information.

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And then there's the
specific, special guys

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with special functions.
 

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And here there'll be sines and
cosines and exponentials.

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Other places in applied math,
there are Bessel functions

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and Legendre functions.
 

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Special guys.
 

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So here, these are special.
 

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And how do I
complete K inverse?

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So this four, three, two, one.
 

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Let me complete T inverse.
 

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You probably know T
inverse already.

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So T, this is, four, three,
two, one, is when the load is

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way over at the far left end
and it's just descending.

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And now I'm going to, let me
show you how I write it in.

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Pay attention here
to the diagonal.

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So this will be three,
three, two, one.

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Do you see that's the solution
that's sort of like this one?

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That's the second column of
the inverse so it's solving,

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I'm solving T times T
inverse equals I here.

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It's the second column
is the guy with a one

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in the second place.
 

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So that's where the load is,
in position number two.

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00:05:42 --> 00:05:46
So I'm level, three,
three up to that load.

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00:05:46 --> 00:05:51
And then I'm dropping
after the load.

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What's the third
column of T inverse?

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I started with that first
column and I knew that the

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answer would be symmetric
because T is symmetric,

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so that allowed me to
write the first row.

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And now we can
fill in the rest.

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So what do you think, if the
point load is now, I'm looking

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at the third column, third
column of the identity, the

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load has moved down to
position number three.

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So what do I have
there and there?

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Two and two.
 

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And what do I have last?
 

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One.
 

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It's dropping to zero.
 

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00:06:28 --> 00:06:31
You could put zero in
green here if you wanted.

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Zero is the unseen last
boundary, you know,

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row at this end.
 

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And finally, what's
happening here?

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What do I get from that?
 

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All one, one, one
to the diagonal.

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And then sure enough
it drops to zero.

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So this would be a case
where the load is there.

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It would be one, one,
one, one and then boom.

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No, it wouldn't be.
 

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00:07:07 --> 00:07:08
It'd be more like this.
 

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One, one, one, one
and then down to.

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That's a pretty clean inverse.
 

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That's a very beautiful matrix.
 

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Don't you admire that matrix?
 

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I mean, if they were all
like that, gee, this

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would be a great world.
 

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But of course it's not sparse.
 

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That's why we don't
often use the inverse.

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Because we had a sparse
matrix T that was really

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fast to compute with.
 

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And here, if you tell me the
inverse, you've actually

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slowed me down.
 

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Because you've given me now a
dense matrix, no zeroes even

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and multiplying T inverse times
the right side would be slower

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than just doing elimination.
 

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Now this is the kind of
more interesting one.

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Because this is the one
that has to go up to the

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diagonal and then down.
 

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So let me, can I fell in what
I the way this one goes?

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I'm going upwards to the
diagonal and then I'm

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coming down to zero.
 

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Remember that I'm coming
down to zero on this K.

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So Zero, zero, zero, zero
is kind of the row number.

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If that's row number zero,
here's one, two, three,

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four, the real thing.
 

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And then row five is getting
back to zero again.

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00:08:48 --> 00:08:52
So what do you think, finish
the rest of that column.

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So you're telling me now
the response to the

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load in position two.
 

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So it's going to
look like this.

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In fact, it's going to
look very like this.

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There's the three and then
this is in position two.

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And then I'm going to have
something here and something

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here and it'll drop to zero.
 

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What do I get?
 

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Four, two.
 

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Six, four, two, zero.
 

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It's dropping to Zero.
 

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I'm going to finish this in but
then I'm going to look back and

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see have I really got it right.
 

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How does this go now?
 

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Two, let's see.
 

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Now it's going up from zero
to two to four to six.

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That's on the diagonal.
 

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Now it starts down.
 

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It's got to get to zero,
so that'll be a three.

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Here is a one going up to
two to three to four.

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Is that right?
 

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And then dropped fast to zero.
 

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Is that correct?
 

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Think so, yep.
 

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Except, wait a minute now.
 

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We've got the right
overall picture.

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Climbing up, dropping down.
 

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Climbing up, dropping down.
 

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Climbing up, dropping down.
 

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All good.
 

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But we haven't yet got, we
haven't checked yet that

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the change in the slope
is supposed to be one.

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And it's not.
 

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Here the slope is like, three,
It's going up by threes and

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then it's going down by twos.
 

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So we've gone from going
up at a slope of three to

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down to a slope of two.
 

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Up three, down just like this.
 

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But that would be a
change in slope of five.

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Therefore there's a 1/5.
 

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So this is going up with a
slope of four and down with a

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slope of one. four dropping to
one when I divide by the

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five that's what I like.
 

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Here is up by twos, down by
threes, again it's a change

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of five so I need the five.
 

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Up by ones, down by four.
 

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Sudden, that's a
fast drop of four.

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Again, the slope changed
by five, dividing by

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five, that's got it.
 

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So that's my picture.
 

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You could now create K
inverse for any size.

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And more than that, sort of
see into K inverse what

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those numbers are.
 

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Because if I wrote the five by
five or six by six doing it

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a column at a time, it would
look like a bunch of numbers.

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00:11:42 --> 00:11:44
But you see it now.
 

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Do you see the pattern?
 

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Right.
 

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This is one way to get to those
inverses, and homework problems

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are offering other ways.
 

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T, in particular, is
quite easy to invert.

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Do I have any other comment on
inverses before the lecture on

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eigenvalues really starts?
 

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Maybe I do have one comment,
one important comment.

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It's this, and I won't
develop it in full,

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but let's just say it.
 

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What if the load is
not a delta function?

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What if I have other loads?
 

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Like the uniform load of all
ones or any other load?

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What if the discrete load
here is not a delta vector?

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I now know the responses
to each column of

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the identity, right?
 

219
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If I put a load in position
one, there's the response.

220
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If I put a load in position
two, there is the response.

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00:12:58 --> 00:13:03
Now, what if I
have other loads?

222
00:13:03 --> 00:13:05
Let me take a typical load.
 

223
00:13:05 --> 00:13:10
What if the load was, well,
the one we looked at before.

224
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If the load was .
 

225
00:13:13 --> 00:13:20
So that I had, the bar
was hanging by its own

226
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weight, let's say.
 

227
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In other words, could I
solve all problems by

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knowing these answers?
 

229
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That's what I'm
trying to get to.

230
00:13:33 --> 00:13:38
If I know these special delta
loads, then can I get the

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solution for every load?
 

232
00:13:41 --> 00:13:42
Yes, no?
 

233
00:13:42 --> 00:13:43
What do you think?
 

234
00:13:43 --> 00:13:45
Yes, right.
 

235
00:13:45 --> 00:13:49
Now with this matrix it's kind
of easy to see because if you

236
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know the inverse matrix, well
you're obviously in business.

237
00:13:53 --> 00:13:59
If I had another load, say
another load f for load, I

238
00:13:59 --> 00:14:03
would just multiply by
K inverse, no problem.

239
00:14:03 --> 00:14:05
But I want to look
a little deeper.

240
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Because if I had other loads
here than a delta function,

241
00:14:11 --> 00:14:15
obviously if I had two delta
functions I could just

242
00:14:15 --> 00:14:17
combine the two solutions.
 

243
00:14:17 --> 00:14:20
That's linearity that
we're using all the time.

244
00:14:20 --> 00:14:23
If I had ten delta functions
I could combine them.

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00:14:23 --> 00:14:29
But then suppose I had instead
of a bunch of spikes, instead

246
00:14:29 --> 00:14:33
of a bunch of point loads,
I had a distributed load.

247
00:14:33 --> 00:14:38
Like all ones, how
could I do it?

248
00:14:38 --> 00:14:39
Main point is I could.
 

249
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Right?
 

250
00:14:40 --> 00:14:44
If I know these answers,
I know all answers.

251
00:14:44 --> 00:14:49
If I know the response to a
load at each point, then-- come

252
00:14:49 --> 00:14:50
back to the discrete one.
 

253
00:14:50 --> 00:14:57
What would be the answer if
the load was ?

254
00:14:57 --> 00:15:06
Suppose I now try to solve
the equation Ku=ones(4,1),

255
00:15:06 --> 00:15:08
so all ones.
 

256
00:15:08 --> 00:15:09
What would be the answer?
 

257
00:15:09 --> 00:15:12
How would I get it?
 

258
00:15:12 --> 00:15:15
I would just add the columns.
 

259
00:15:15 --> 00:15:20
Now why would I do that?
 

260
00:15:20 --> 00:15:21
Right.
 

261
00:15:21 --> 00:15:27
Because this, the right-hand
side, the input is the sum of

262
00:15:27 --> 00:15:31
the four columns, the
four special inputs.

263
00:15:31 --> 00:15:36
So the output is the sum of
the four outputs, right.

264
00:15:36 --> 00:15:39
In other words, as you saw,
we must know everything.

265
00:15:39 --> 00:15:41
And that's the way
we really know it.

266
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By linearity.
 

267
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If the input is a combination
of these, the output is the

268
00:15:47 --> 00:15:49
same combination of those.
 

269
00:15:49 --> 00:15:50
Right.
 

270
00:15:50 --> 00:15:55
So, for example, in this T
case, if input was, if I

271
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did Tu=ones, I would just
add those and the output

272
00:16:02 --> 00:16:06
would be .
 

273
00:16:06 --> 00:16:10
That would be the output
from .

274
00:16:10 --> 00:16:20
And now, oh boy.
 

275
00:16:20 --> 00:16:26
Actually, let me just introduce
a guy's name for these

276
00:16:26 --> 00:16:31
solutions and not
today show you.

277
00:16:31 --> 00:16:33
You have the idea, of course.
 

278
00:16:33 --> 00:16:37
Here we added because
everything was discrete.

279
00:16:37 --> 00:16:40
So you know what we're
going to do over here.

280
00:16:40 --> 00:16:44
We'll take integrals , right?
 

281
00:16:44 --> 00:16:51
A general load will be an
integral over point loads.

282
00:16:51 --> 00:16:53
That's the idea.
 

283
00:16:53 --> 00:16:54
A fundamental idea.
 

284
00:16:54 --> 00:17:00
That some other load, f(x) is
an integral of these guys.

285
00:17:00 --> 00:17:05
So the solution will be the
same integral of these guys.

286
00:17:05 --> 00:17:08
Let me not go there except to
tell you the name, because

287
00:17:08 --> 00:17:11
it's a very famous name.
 

288
00:17:11 --> 00:17:16
This solution u with the
delta function is called

289
00:17:16 --> 00:17:17
the Green's function.
 

290
00:17:17 --> 00:17:20
So I've now introduced
the idea, this is the

291
00:17:20 --> 00:17:21
Green's function.
 

292
00:17:21 --> 00:17:25
This guy is the Green's
function for the

293
00:17:25 --> 00:17:30
fixed-fixed problem.
 

294
00:17:30 --> 00:17:33
And this guy is the
Green's function for

295
00:17:33 --> 00:17:36
the free-fixed problem.
 

296
00:17:36 --> 00:17:40
And the whole point is, maybe
this is the one point I want

297
00:17:40 --> 00:17:44
you to sort of see
always by analogy.

298
00:17:44 --> 00:17:50
The Green's function is
just like the inverse.

299
00:17:50 --> 00:17:52
What is the Green's function?
 

300
00:17:52 --> 00:17:59
The Green's function is the
response at x to the u(x)

301
00:17:59 --> 00:18:03
when the input, when
the impulses is at a.

302
00:18:03 --> 00:18:04
So it sort of depends
on two things.

303
00:18:04 --> 00:18:09
It depends on the position a
of the input and it tells you

304
00:18:09 --> 00:18:14
the response at position x.
 

305
00:18:14 --> 00:18:19
And often we would use the
letter G for Green's.

306
00:18:19 --> 00:18:22
So it depends on x(a).
 

307
00:18:23 --> 00:18:30
And maybe I'm happy if you just
sort of see in some way what we

308
00:18:30 --> 00:18:33
did there is just like
what we did here.

309
00:18:33 --> 00:18:37
And therefore the Green's
function must be just a

310
00:18:37 --> 00:18:46
differential, continuous
version of an inverse matrix.

311
00:18:46 --> 00:18:54
Let's move on to eigenvalues
with that point sort of made,

312
00:18:54 --> 00:18:59
but not driven home by
many, many examples.

313
00:18:59 --> 00:19:15
Question, I'll take
a question, shoot.

314
00:19:15 --> 00:19:21
Why did I increase zero, three,
six and then decrease six?

315
00:19:21 --> 00:19:29
Well intuitively it's because
this is copying this.

316
00:19:29 --> 00:19:32
What's wonderful is that
it's a perfect copy.

317
00:19:32 --> 00:19:36
I mean, intuitively the
solution to our difference

318
00:19:36 --> 00:19:39
equation should be like
the solution to our

319
00:19:39 --> 00:19:40
differential equation.
 

320
00:19:40 --> 00:19:44
That's why if we have some
computational, some

321
00:19:44 --> 00:19:47
differential equation that we
can't solve, which would be

322
00:19:47 --> 00:19:51
much more typical than this
one, that we couldn't solve it

323
00:19:51 --> 00:19:57
exactly by pencil and paper, we
would replace derivatives by

324
00:19:57 --> 00:20:00
differences and go over here
and we would hope that they

325
00:20:00 --> 00:20:02
were like pretty close.
 

326
00:20:02 --> 00:20:07
Here they're right,
they're the same.

327
00:20:07 --> 00:20:08
Oh the other columns?
 

328
00:20:08 --> 00:20:09
Absolutely.
 

329
00:20:09 --> 00:20:11
These guys?
 

330
00:20:11 --> 00:20:14
Zero, two, four, six going up.
 

331
00:20:14 --> 00:20:18
Six, three, zero coming back.
 

332
00:20:18 --> 00:20:25
So that's a discrete
thing of one like that.

333
00:20:25 --> 00:20:29
And then the next guy and the
last guy would be going up

334
00:20:29 --> 00:20:34
one, two, three, four
and then sudden drop.

335
00:20:34 --> 00:20:35
Thanks for all questions.
 

336
00:20:35 --> 00:20:40
I mean, this sort of, by adding
these guys in, the first one

337
00:20:40 --> 00:20:41
actually went up that way.
 

338
00:20:41 --> 00:20:45
You see the Green's functions.
 

339
00:20:45 --> 00:20:48
But of course this has a
Green's function for every

340
00:20:48 --> 00:20:53
a. x and a are running all
the way from zero to one.

341
00:20:53 --> 00:20:58
Here they're just
discrete positions.

342
00:20:58 --> 00:21:02
Thanks.
 

343
00:21:02 --> 00:21:07
So playing with these delta
functions and coming up with

344
00:21:07 --> 00:21:12
this solution, well, as I say,
different ways to do it.

345
00:21:12 --> 00:21:16
I worked through one way
in class last time.

346
00:21:16 --> 00:21:18
It takes practice.
 

347
00:21:18 --> 00:21:21
So that's what the
homework's really for.

348
00:21:21 --> 00:21:26
You can see me come up with
this thing, then you can, with

349
00:21:26 --> 00:21:29
leisure, you can follow the
steps, but you've gotta

350
00:21:29 --> 00:21:32
do it yourself to see.
 

351
00:21:32 --> 00:21:36
Eigenvalues and, of
course, eigenvectors.

352
00:21:36 --> 00:21:46
We have to give
them a fair shot.

353
00:21:46 --> 00:21:49
Square matrix.
 

354
00:21:49 --> 00:21:54
So I'm talking about general,
what eigenvectors and

355
00:21:54 --> 00:21:57
eigenvalues are and
why do we want them.

356
00:21:57 --> 00:22:01
I'm always trying to say
what's the purpose, you know,

357
00:22:01 --> 00:22:07
not doing this just for
abstract linear algebra.

358
00:22:07 --> 00:22:11
We do this, we look for
these things because they

359
00:22:11 --> 00:22:16
tremendously simplify a
problem if we can find it.

360
00:22:16 --> 00:22:19
So what's an eigenvector?
 

361
00:22:19 --> 00:22:24
The eigenvalue is this number,
lambda, and the eigenvector

362
00:22:24 --> 00:22:26
is this vector y.
 

363
00:22:26 --> 00:22:33
And now, how do I
think about those?

364
00:22:33 --> 00:22:37
Suppose I take a vector
and I multiply by A.

365
00:22:37 --> 00:22:42
So the vector is headed
off in some direction.

366
00:22:42 --> 00:22:44
Here's a vector v.
 

367
00:22:44 --> 00:22:47
If I multiply, and I'm given
this matrix, so I'm given the

368
00:22:47 --> 00:22:51
matrix, whatever my matrix is.
 

369
00:22:51 --> 00:22:54
Could be one of those
matrices, any other matrix.

370
00:22:54 --> 00:23:00
If I multiply that by v,
I get some result, Av.

371
00:23:00 --> 00:23:01
What do I do?
 

372
00:23:01 --> 00:23:06
I look at that and I say that
v was not an eigenvector.

373
00:23:06 --> 00:23:11
Eigenvectors are the special
vectors which come out

374
00:23:11 --> 00:23:12
in the same direction.
 

375
00:23:12 --> 00:23:15
Av comes out parallel to v.
 

376
00:23:15 --> 00:23:18
So this was not an eigenvector.
 

377
00:23:18 --> 00:23:21
Very few vectors are
eigenvectors, they're

378
00:23:21 --> 00:23:22
very special.
 

379
00:23:22 --> 00:23:25
Most vectors, that'll
be a typical picture.

380
00:23:25 --> 00:23:33
But there's a few of them
where I've a vector y

381
00:23:33 --> 00:23:35
and I multiply by A.
 

382
00:23:35 --> 00:23:36
And then what's the point?
 

383
00:23:36 --> 00:23:42
Ay is in the same direction.
 

384
00:23:42 --> 00:23:45
It's on that same line as y.
 

385
00:23:45 --> 00:23:48
It could be, it might
be twice as far out.

386
00:23:48 --> 00:23:49
That would be Ay=2y.
 

387
00:23:51 --> 00:23:53
It might go backwards.
 

388
00:23:53 --> 00:23:56
This would be a
possibility, Ay=-y.

389
00:23:56 --> 00:23:58
 
 

390
00:23:58 --> 00:24:02
It could be just halfway.
 

391
00:24:02 --> 00:24:05
It could be, not move at all.
 

392
00:24:05 --> 00:24:06
That's even a possibility.
 

393
00:24:06 --> 00:24:06
Ay=0y.
 

394
00:24:07 --> 00:24:10
Count that.
 

395
00:24:10 --> 00:24:17
Those y's or eigenvectors and
the eigenvalue is just, from

396
00:24:17 --> 00:24:19
this point of view, the
eigenvalue has come in second

397
00:24:19 --> 00:24:24
because it's, so y was a
special vector that

398
00:24:24 --> 00:24:26
kept its direction.
 

399
00:24:26 --> 00:24:32
And then lambda is just the
number, the two, the zero, the

400
00:24:32 --> 00:24:39
minus one, the 1/2 that tells
you stretching, shrinking,

401
00:24:39 --> 00:24:41
reversing, whatever.
 

402
00:24:41 --> 00:24:42
That's the number.
 

403
00:24:42 --> 00:24:45
But y is the vector.
 

404
00:24:45 --> 00:24:54
And notice that if I knew y and
I knew it was an eigenvector,

405
00:24:54 --> 00:24:59
then of course if I multiply by
A, I'll learn the eigenvalue.

406
00:24:59 --> 00:25:02
And if I knew an eigenvalue,
you'll see how I could

407
00:25:02 --> 00:25:03
find the eigenvector.
 

408
00:25:03 --> 00:25:06
Problem is you have
to find them both.

409
00:25:06 --> 00:25:07
And they multiply each other.
 

410
00:25:07 --> 00:25:11
So we're not talking about
linear equations anymore.

411
00:25:11 --> 00:25:13
Because one unknown is
multiplying another.

412
00:25:13 --> 00:25:19
But we'll find a way to look
to discover eigenvectors

413
00:25:19 --> 00:25:23
and eigenvalues.
 

414
00:25:23 --> 00:25:27
I said I would try to make
clear what's the purpose.

415
00:25:27 --> 00:25:36
The purpose is that in this
direction on this y line, line

416
00:25:36 --> 00:25:43
of multiples of yA is just
acting like a number.

417
00:25:43 --> 00:25:48
A is some big n by n,
1,000 by 1,000 matrix.

418
00:25:48 --> 00:25:50
So a million numbers.
 

419
00:25:50 --> 00:25:59
But on this line if we find an
eigenline you could say, an

420
00:25:59 --> 00:26:03
eigendirection in that
direction, all the

421
00:26:03 --> 00:26:06
complications of A are gone.
 

422
00:26:06 --> 00:26:08
It's just acting like a number.
 

423
00:26:08 --> 00:26:14
So in particular we could solve
1,000 differential equations

424
00:26:14 --> 00:26:23
with 1,000 unknown u's with
this 1,000 by 1,000 matrix.

425
00:26:23 --> 00:26:27
We can find a solution
and this is where the

426
00:26:27 --> 00:26:34
eigenvector eigenvalue
are going to pay off.

427
00:26:34 --> 00:26:35
You recognize this.
 

428
00:26:35 --> 00:26:38
Matrix A is of size 1,000.
 

429
00:26:38 --> 00:26:41
And u is a vector
of 1,000 unknowns.

430
00:26:41 --> 00:26:44
So that's a system
of 1,000 equations.

431
00:26:44 --> 00:26:50
But if we have found an
eigenvector and it's eigenvalue

432
00:26:50 --> 00:26:56
then the equation will, if it
starts in that direction it'll

433
00:26:56 --> 00:26:59
stay in that direction and
the matrix will just be

434
00:26:59 --> 00:27:01
acting like a number.
 

435
00:27:01 --> 00:27:03
And we know how to
solve U'=lambda*u.

436
00:27:03 --> 00:27:06
 
 

437
00:27:06 --> 00:27:10
That one by one scalar problem
we know how to solve.

438
00:27:10 --> 00:27:13
The solution to that
is e to the lambda*t.

439
00:27:13 --> 00:27:17
 
 

440
00:27:17 --> 00:27:21
And of course it could
have a constant do that.

441
00:27:21 --> 00:27:25
Don't forget that these
equations are linear.

442
00:27:25 --> 00:27:30
If I multiply it, if I take
2e^(lambda*t), I have a two

443
00:27:30 --> 00:27:32
here and a two here and
it's just as good.

444
00:27:32 --> 00:27:37
So I better allow that as well.
 

445
00:27:37 --> 00:27:39
A constant times e^(lambda*t)y.
 

446
00:27:41 --> 00:27:43
Notice this is a vector.
 

447
00:27:43 --> 00:27:47
It's a number times a
number, the growth.

448
00:27:47 --> 00:27:50
So the lambda is now, for the
differential equation, the

449
00:27:50 --> 00:27:54
lambda, this number
lambda is crucial.

450
00:27:54 --> 00:27:58
It's telling us whether the
solution grows, whether it

451
00:27:58 --> 00:28:01
decays, whether it oscillates.
 

452
00:28:01 --> 00:28:05
And we're just looking at this
one normal mode, you could say

453
00:28:05 --> 00:28:09
normal mode for eigenvector y.
 

454
00:28:09 --> 00:28:18
We certainly have not found
all possible solutions.

455
00:28:18 --> 00:28:25
If we have an eigenvector,
we found that one.

456
00:28:25 --> 00:28:30
And there's other uses
and then, let me think.

457
00:28:30 --> 00:28:31
Other uses, what?
 

458
00:28:31 --> 00:28:34
So let me write again the
fundamental equation,

459
00:28:34 --> 00:28:34
Ay=lambda*y.
 

460
00:28:34 --> 00:28:37
 
 

461
00:28:37 --> 00:28:41
So that was a
differential equation.

462
00:28:41 --> 00:28:43
Going forward in time.
 

463
00:28:43 --> 00:28:48
Now if we go forward in
steps we might multiply

464
00:28:48 --> 00:28:55
by A at every step.
 

465
00:28:55 --> 00:28:59
Tell me an eigenvector
of A squared.

466
00:28:59 --> 00:29:01
I'm looking for a vector that
doesn't change direction

467
00:29:01 --> 00:29:05
when I multiply twice by A.
 

468
00:29:05 --> 00:29:10
You're going to tell me
it's y. y will work.

469
00:29:10 --> 00:29:15
If I multiply once by A
I get lambda times y.

470
00:29:15 --> 00:29:20
When I multiply again by A I
get lambda squared times y.

471
00:29:20 --> 00:29:30
You see eigenvalues are great
for powers of a matrix, for

472
00:29:30 --> 00:29:33
differential equations.
 

473
00:29:33 --> 00:29:37
The nth power will just take
the eigenvalue to the nth.

474
00:29:37 --> 00:29:42
The nth power of A will just
have lambda to the nth there.

475
00:29:42 --> 00:29:47
You know, the pivots of
a matrix are all messed

476
00:29:47 --> 00:29:49
up when I square it.
 

477
00:29:49 --> 00:29:52
I can't see what's
happening with the pivots.

478
00:29:52 --> 00:29:56
The eigenvalues are a different
way to look at a matrix.

479
00:29:56 --> 00:30:01
The pivots are critical numbers
for steady-state problems.

480
00:30:01 --> 00:30:07
The eigenvalues are the
critical numbers for moving

481
00:30:07 --> 00:30:11
problems, dynamic problems,
things are oscillating

482
00:30:11 --> 00:30:13
or growing or decaying.
 

483
00:30:13 --> 00:30:21
And by the way, let's just
recognize since this is the

484
00:30:21 --> 00:30:29
only thing that's changing in
time, what would be the, I'll

485
00:30:29 --> 00:30:30
just go down here,
e^(lambda*t).

486
00:30:31 --> 00:30:32
Let's just look and see.
 

487
00:30:32 --> 00:30:35
When would I have decay?
 

488
00:30:35 --> 00:30:38
Which you might want
to call stability.

489
00:30:38 --> 00:30:40
A stable problem.
 

490
00:30:40 --> 00:30:43
What would be the condition
on lambda to get

491
00:30:43 --> 00:30:46
for this to decay.
 

492
00:30:46 --> 00:30:49
Lambda less than zero.
 

493
00:30:49 --> 00:30:52
Now there's one little
bit of bad news.

494
00:30:52 --> 00:30:55
Lambda could be complex.
 

495
00:30:55 --> 00:30:58
Lambda could be 3+4i.
 

496
00:30:58 --> 00:31:00
 
 

497
00:31:00 --> 00:31:03
It can be a complex
number, these eigenvalues

498
00:31:03 --> 00:31:09
even if A is real.
 

499
00:31:09 --> 00:31:11
You'll say, how'd that
happen, let me see?

500
00:31:11 --> 00:31:13
I didn't think.
 

501
00:31:13 --> 00:31:14
Well, let me finish
this thought.

502
00:31:14 --> 00:31:18
Suppose lambda was 3+4i.
 

503
00:31:18 --> 00:31:22
 
 

504
00:31:22 --> 00:31:25
So I'm thinking about what
would either the lambda*t

505
00:31:25 --> 00:31:27
do in that case?
 

506
00:31:27 --> 00:31:30
So this is small example.
 

507
00:31:30 --> 00:31:32
If I had lambda (3+4i)t.
 

508
00:31:32 --> 00:31:35
 
 

509
00:31:35 --> 00:31:40
What does that do
as time grows?

510
00:31:40 --> 00:31:42
It's going to grow
and oscillate.

511
00:31:42 --> 00:31:45
And what decides the growth?
 

512
00:31:45 --> 00:31:46
The real part.
 

513
00:31:46 --> 00:31:50
So it's really the decay
or growth is decided

514
00:31:50 --> 00:31:51
by the real part.
 

515
00:31:51 --> 00:31:55
The three, either the 3t,
that would be a growth.

516
00:31:55 --> 00:31:58
Let me put growth.
 

517
00:31:58 --> 00:32:01
And that would be, of
course, unstable.

518
00:32:01 --> 00:32:05
And that's a problem when
I have a real part of

519
00:32:05 --> 00:32:07
lambda bigger than zero.
 

520
00:32:07 --> 00:32:13
And then if lambda has a zero
real part, so it's pure

521
00:32:13 --> 00:32:17
oscillation, let me just
take a case like that.

522
00:32:17 --> 00:32:18
So e^(4it).
 

523
00:32:18 --> 00:32:20
 
 

524
00:32:20 --> 00:32:24
So that would be,
oscillating, right?

525
00:32:24 --> 00:32:31
It's cos(4t) + i*sin(4t),
it's just oscillating.

526
00:32:31 --> 00:32:39
So in this discussion we've
seen growth and decay.

527
00:32:39 --> 00:32:41
Tell me that parallels
because I'm always shooting

528
00:32:41 --> 00:32:43
for the parallels.
 

529
00:32:43 --> 00:32:45
What about the growth of A?
 

530
00:32:45 --> 00:32:52
What matrices, how can
I recognize a matrix

531
00:32:52 --> 00:32:56
whose powers grow?
 

532
00:32:56 --> 00:33:02
How can I recognize a matrix
whose powers go to zero?

533
00:33:02 --> 00:33:06
I'm asking you for powers
here, over there for

534
00:33:06 --> 00:33:08
exponentials somehow.
 

535
00:33:08 --> 00:33:16
So here would be A to higher
and higher powers goes to

536
00:33:16 --> 00:33:18
zero, the zero matrix.
 

537
00:33:18 --> 00:33:21
In other words, when I
multiply, multiply, multiply by

538
00:33:21 --> 00:33:25
that matrix I get smaller and
smaller and smaller matrices,

539
00:33:25 --> 00:33:26
zero in the limit.
 

540
00:33:26 --> 00:33:33
What do you think's the
test on the lambda now?

541
00:33:33 --> 00:33:37
So what are the eigenvalues
of A to the K?

542
00:33:37 --> 00:33:38
Let's see.
 

543
00:33:38 --> 00:33:41
If A had eigenvalues lambda, A
squared will have eigenvalues

544
00:33:41 --> 00:33:45
lambda squared, A cubed will
have eigenvalues lambda cubed,

545
00:33:45 --> 00:33:48
A to the thousandth will have
eigenvalues lambda

546
00:33:48 --> 00:33:49
to the thousandth.
 

547
00:33:49 --> 00:33:54
And what's the test for
that to be getting small?

548
00:33:54 --> 00:33:58
Lambda less than one.
 

549
00:33:58 --> 00:34:03
So the test for stability will
be in the discrete case.

550
00:34:03 --> 00:34:08
It won't be the real part of
lambda, it'll be the size

551
00:34:08 --> 00:34:10
of lambda less than one.
 

552
00:34:10 --> 00:34:16
And growth would be the size
of lambda greater than one.

553
00:34:16 --> 00:34:20
And again, there'd be this
borderline case when

554
00:34:20 --> 00:34:24
the eigenvalue has
magnitude exactly one.

555
00:34:24 --> 00:34:31
So you're seeing here and also
here the idea that we may have

556
00:34:31 --> 00:34:34
to deal with complex
numbers here.

557
00:34:34 --> 00:34:38
We don't have to deal with
the whole world of complex

558
00:34:38 --> 00:34:42
functions and everything but
it's possible for complex

559
00:34:42 --> 00:34:45
numbers to come in.
 

560
00:34:45 --> 00:34:49
Well while I'm saying that,
why don't I give an example

561
00:34:49 --> 00:34:58
where it would come in.
 

562
00:34:58 --> 00:35:03
This is going to be
a real matrix with

563
00:35:03 --> 00:35:07
complex eigenvalues.
 

564
00:35:07 --> 00:35:11
Complex lambdas.
 

565
00:35:11 --> 00:35:19
It'll be an example.
 

566
00:35:19 --> 00:35:26
So I guess I'm looking for a
matrix where y and Ay never

567
00:35:26 --> 00:35:29
come out in the same direction.
 

568
00:35:29 --> 00:35:34
For real y's I know, okay,
here's a good matrix.

569
00:35:34 --> 00:35:41
Take the matrix that rotates
every vector by 90 degrees.

570
00:35:41 --> 00:35:43
Or by theta.
 

571
00:35:43 --> 00:35:47
But let's say here's a
matrix that rotates every

572
00:35:47 --> 00:35:55
vector by 90 degrees.
 

573
00:35:55 --> 00:35:57
I'm going to raise this
board and hide it behind

574
00:35:57 --> 00:35:58
there in a minute.
 

575
00:35:58 --> 00:36:05
I just wanted to just to open
up this thought that we will

576
00:36:05 --> 00:36:09
have to face complex numbers.
 

577
00:36:09 --> 00:36:12
If you know how to multiply
two complex numbers and

578
00:36:12 --> 00:36:16
add them, you're ok.
 

579
00:36:16 --> 00:36:20
This isn't going to
turn into a big deal.

580
00:36:20 --> 00:36:25
But let's just realize that
suppose that matrix, if I put

581
00:36:25 --> 00:36:30
in a vector y and I multiply
by that matrix, it'll turn

582
00:36:30 --> 00:36:33
it through 90 degrees.
 

583
00:36:33 --> 00:36:35
So y couldn't be
an eigenvector.

584
00:36:35 --> 00:36:37
That's the point I'm
trying to make.

585
00:36:37 --> 00:36:42
No real vector could be the
eigenvector of a rotation

586
00:36:42 --> 00:36:46
matrix because every
vector gets turned.

587
00:36:46 --> 00:36:52
So that's an example where
you'd have to go to complex

588
00:36:52 --> 00:37:00
vectors. and I think if I tried
the vector 1i, so I'm letting

589
00:37:00 --> 00:37:03
the square root of minus one
into here, then I think

590
00:37:03 --> 00:37:05
it would come out.
 

591
00:37:05 --> 00:37:09
If I do that multiplication
I get minus i.

592
00:37:09 --> 00:37:10
And I get one.
 

593
00:37:10 --> 00:37:15
And I think that this
is, what is it?

594
00:37:15 --> 00:37:17
This is probably
minus i times that.

595
00:37:17 --> 00:37:32
So this is minus i
times the input.

596
00:37:32 --> 00:37:34
No big deal.
 

597
00:37:34 --> 00:37:36
That was like, you
can forget that.

598
00:37:36 --> 00:37:43
It's just complex
numbers can come in.

599
00:37:43 --> 00:37:52
Now let me come back to the
main point about eigenvectors.

600
00:37:52 --> 00:37:57
Things can be complex.
 

601
00:37:57 --> 00:38:02
So the main point is
how do we use them?

602
00:38:02 --> 00:38:08
And how many are there?
 

603
00:38:08 --> 00:38:10
Here's the key.
 

604
00:38:10 --> 00:38:14
A typical, good matrix which
includes every symmetric

605
00:38:14 --> 00:38:19
matrix, so it includes all of
our examples and more, if it's

606
00:38:19 --> 00:38:24
of size 1,000, it will have
1,000 different eigenvectors.

607
00:38:24 --> 00:38:29
And let me just say for our
symmetric matrices those

608
00:38:29 --> 00:38:33
eigenvectors will all be real.
 

609
00:38:33 --> 00:38:37
They're great, the eigenvectors
of symmetric matrices.

610
00:38:37 --> 00:38:40
Oh, let me find them for one
particular symmetric matrix.

611
00:38:40 --> 00:38:47
Say this guy.
 

612
00:38:47 --> 00:38:49
So that's a matrix. two by two.
 

613
00:38:49 --> 00:38:53
How many eigenvectors
am I now looking for?

614
00:38:53 --> 00:38:55
Two.
 

615
00:38:55 --> 00:38:59
You could say, how
do I find them?

616
00:38:59 --> 00:39:07
Maybe with a two by two,
I can even just wing it.

617
00:39:07 --> 00:39:13
We can come up with a vector
that is an eigenvector.

618
00:39:13 --> 00:39:18
Actually that's what we're
going to do here is we're going

619
00:39:18 --> 00:39:21
to guess the eigenvectors and
then we're going to show that

620
00:39:21 --> 00:39:24
they really are eigenvectors
and then we'll know the

621
00:39:24 --> 00:39:27
eigenvalues and it's fantastic.
 

622
00:39:27 --> 00:39:31
So like let's start here
with the two by two case.

623
00:39:31 --> 00:39:33
Anybody spot an eigenvector?
 

624
00:39:33 --> 00:39:35
Is  an eigenvector?
 

625
00:39:35 --> 00:39:36
Try .
 

626
00:39:36 --> 00:39:39
What comes out of ?
 

627
00:39:39 --> 00:39:43
Well that picks the
first column, right?

628
00:39:43 --> 00:39:45
That's how I see
multiplying by .

629
00:39:45 --> 00:39:48
That says take one of
the first column.

630
00:39:48 --> 00:39:52
And is it an eigenvector?
 

631
00:39:52 --> 00:39:53
Yes, no?
 

632
00:39:53 --> 00:39:55
No.
 

633
00:39:55 --> 00:39:59
This vector is not in the
same direction as that one.

634
00:39:59 --> 00:40:00
No good.
 

635
00:40:00 --> 00:40:09
Now can you tell
me one that is?

636
00:40:09 --> 00:40:15
You're going to guess
it. . try .

637
00:40:15 --> 00:40:21
Do the multiplication
and what do you get?

638
00:40:21 --> 00:40:23
Right?
 

639
00:40:23 --> 00:40:29
If I input this vector
y, what do I get out?

640
00:40:29 --> 00:40:33
Actually I get y itself.
 

641
00:40:33 --> 00:40:36
Right?
 

642
00:40:36 --> 00:40:39
The point is it didn't change
direction, and it didn't

643
00:40:39 --> 00:40:40
even change length.
 

644
00:40:40 --> 00:40:42
So what's the
eigenvalue for that?

645
00:40:42 --> 00:40:47
So I've got one eigenvalue
now, one eigenvector. .

646
00:40:47 --> 00:40:51
And I've got the eigenvalue.
 

647
00:40:51 --> 00:40:53
So here are the
vectors, the y's.

648
00:40:53 --> 00:40:55
And here are the lambdas.
 

649
00:40:55 --> 00:41:01
And I've got one of them
and it's one, right?

650
00:41:01 --> 00:41:03
Would you like to
guess the other one?

651
00:41:03 --> 00:41:05
I'm only looking for
two because it's a

652
00:41:05 --> 00:41:06
two by two matrix.
 

653
00:41:06 --> 00:41:10
So let me erase here, hope
that you'll come up with

654
00:41:10 --> 00:41:17
another one.  is
certainly worth a try.

655
00:41:17 --> 00:41:19
Let's test it.
 

656
00:41:19 --> 00:41:21
If it's an eigenvector,
then it should come out

657
00:41:21 --> 00:41:22
in the same direction.
 

658
00:41:22 --> 00:41:26
What do I get when I do that?
 

659
00:41:26 --> 00:41:28
So I do that multiplication.
 

660
00:41:28 --> 00:41:33
Three and I get three and
minus three, so have

661
00:41:33 --> 00:41:35
we got an eigenvector?
 

662
00:41:35 --> 00:41:37
Yep.
 

663
00:41:37 --> 00:41:42
And what's, so if this was
y, what is this vector?

664
00:41:42 --> 00:41:42
3y.
 

665
00:41:44 --> 00:41:47
So there's the other
eigenvector is  and the

666
00:41:47 --> 00:41:56
other eigenvalue is three.
 

667
00:41:56 --> 00:42:01
So we did it by
spotting it here.

668
00:42:01 --> 00:42:03
MATLAB can't do it that way.
 

669
00:42:03 --> 00:42:06
It's got to figure it out.
 

670
00:42:06 --> 00:42:12
But we're ahead of
MATLAB this time.

671
00:42:12 --> 00:42:15
So what do I notice?
 

672
00:42:15 --> 00:42:17
What do I notice
about this matrix?

673
00:42:17 --> 00:42:20
It was symmetric.
 

674
00:42:20 --> 00:42:25
And what do I notice
about the eigenvectors?

675
00:42:25 --> 00:42:29
If I show you those two
vectors,  and ,

676
00:42:29 --> 00:42:32
what do you see there?
 

677
00:42:32 --> 00:42:38
They're orthogonal. 
is orthogonal to ,

678
00:42:38 --> 00:42:40
perpendicular is the
same as orthogonal.

679
00:42:40 --> 00:42:49
These are orthogonal,
perpendicular.

680
00:42:49 --> 00:42:54
I can draw them, of course and
see that.  will go, if

681
00:42:54 --> 00:42:58
this is one, it'll go here.
 

682
00:42:58 --> 00:43:00
So that's .
 

683
00:43:00 --> 00:43:03
And  will go there,
it'll go down, this would

684
00:43:03 --> 00:43:06
be the other one. .
 

685
00:43:06 --> 00:43:06
So there's y_1.
 

686
00:43:07 --> 00:43:08
There's y_2.
 

687
00:43:08 --> 00:43:11
And they are perpendicular.
 

688
00:43:11 --> 00:43:17
But of course I don't draw
pictures all the time.

689
00:43:17 --> 00:43:21
What's the test for two
vectors being orthogonal?

690
00:43:21 --> 00:43:23
The dot product.
 

691
00:43:23 --> 00:43:24
The dot product.
 

692
00:43:24 --> 00:43:29
The inner product. y transpose,
y_1 transpose * y_2.

693
00:43:31 --> 00:43:35
Do you prefer to write it
as y_1 with a dot, y_2?

694
00:43:35 --> 00:43:38
 
 

695
00:43:38 --> 00:43:42
This is maybe better because
it's matrix notation.

696
00:43:42 --> 00:43:51
And the point is orthogonal,
the dot product is zero.

697
00:43:51 --> 00:43:53
So that's good.
 

698
00:43:53 --> 00:43:56
Very good, in fact.
 

699
00:43:56 --> 00:43:59
So here's a very
important fact.

700
00:43:59 --> 00:44:06
Symmetric matrices have
orthogonal eigenvectors.

701
00:44:06 --> 00:44:09
What I'm trying to say is
eigenvectors and eigenvalues

702
00:44:09 --> 00:44:13
are like a new way
to look at a matrix.

703
00:44:13 --> 00:44:16
A new way to see into it.
 

704
00:44:16 --> 00:44:22
And when the matrix is
symmetric, what we see is

705
00:44:22 --> 00:44:25
perpendicular eigenvectors.
 

706
00:44:25 --> 00:44:28
And what comment do you have
about the eigenvalues of

707
00:44:28 --> 00:44:32
this symmetric matrix?
 

708
00:44:32 --> 00:44:37
Remembering what was
on the board for this

709
00:44:37 --> 00:44:40
anti-symmetric matrix.
 

710
00:44:40 --> 00:44:44
What was the point about
that anti-symmetric matrix?

711
00:44:44 --> 00:44:51
It's eigenvalues were imaginary
actually, an i there.

712
00:44:51 --> 00:44:53
Here is the opposite.
 

713
00:44:53 --> 00:44:57
What's the property of the
eigenvalues for a symmetric

714
00:44:57 --> 00:45:00
matrix that you just guess?
 

715
00:45:00 --> 00:45:02
They're real.
 

716
00:45:02 --> 00:45:03
They're real.
 

717
00:45:03 --> 00:45:07
Symmetric matrices are great
because they have real

718
00:45:07 --> 00:45:19
eigenvalues and they have
perpendicular eigenvectors and

719
00:45:19 --> 00:45:22
actually, probably if a matrix
has real eigenvalues and

720
00:45:22 --> 00:45:27
perpendicular eigenvectors,
it's going to be symmetric.

721
00:45:27 --> 00:45:32
So symmetry is a great property
and it shows up in a great way

722
00:45:32 --> 00:45:38
in this real eigenvalue, real
lambdas, and orthogonal y's

723
00:45:38 --> 00:45:48
shows up perfectly in
the eigenpicture.

724
00:45:48 --> 00:45:53
Here's a handy little check
on the eigenvalues to

725
00:45:53 --> 00:45:55
see if we got it right.
 

726
00:45:55 --> 00:45:56
Course we did.
 

727
00:45:56 --> 00:45:59
That's one and
three we can get.

728
00:45:59 --> 00:46:03
But let me just show you two
useful checks if you haven't

729
00:46:03 --> 00:46:06
seen eigenvalues before.
 

730
00:46:06 --> 00:46:10
If I add the eigenvalues,
what do I get?

731
00:46:10 --> 00:46:12
Four.
 

732
00:46:12 --> 00:46:15
And I compare that
with adding down the

733
00:46:15 --> 00:46:17
diagonal of the matrix.
 

734
00:46:17 --> 00:46:19
Two and two, four.
 

735
00:46:19 --> 00:46:21
And that check always works.
 

736
00:46:21 --> 00:46:25
The sum of the eigenvalues
matches the sum

737
00:46:25 --> 00:46:26
down the diagonal.
 

738
00:46:26 --> 00:46:30
So that's like, if you got all
the eigenvalues but one, that

739
00:46:30 --> 00:46:32
would tell you the last one.
 

740
00:46:32 --> 00:46:36
Because the sum of the
eigenvalues matches the

741
00:46:36 --> 00:46:39
sum down the diagonal.
 

742
00:46:39 --> 00:46:45
You have no clue where that
comes from but it's true.

743
00:46:45 --> 00:46:48
And another useful fact.
 

744
00:46:48 --> 00:46:52
If I multiply the
eigenvalues what do I get?

745
00:46:52 --> 00:46:53
Three?
 

746
00:46:53 --> 00:46:58
And now, where do you
see a three over here?

747
00:46:58 --> 00:47:00
The determinant.
 

748
00:47:00 --> 00:47:01
4-1=3.
 

749
00:47:03 --> 00:47:07
Can I just write those two
facts with no idea of proof.

750
00:47:07 --> 00:47:17
The sum of the lambdas, I could
write "sum" this is for any

751
00:47:17 --> 00:47:22
matrix, the sum of the lambdas
is equal to the, it's called

752
00:47:22 --> 00:47:25
the trace of the matrix.
 

753
00:47:25 --> 00:47:29
The trace of the matrix is
the sum down the diagonal.

754
00:47:29 --> 00:47:37
And the product of the lambdas,
lambda_1 times lambda_2 is the

755
00:47:37 --> 00:47:40
determinant of the matrix.
 

756
00:47:40 --> 00:47:43
Or if I had ten eigenvalues,
I would multiply all ten and

757
00:47:43 --> 00:47:47
I'd get the determinant.
 

758
00:47:47 --> 00:47:51
So that's some facts
about eigenvalues.

759
00:47:51 --> 00:47:56
There's more, of course, in
section 1.5 about how you

760
00:47:56 --> 00:48:04
would find eigenvalues and
how you would use them.

761
00:48:04 --> 00:48:09
That's of course the key point,
is how would we use them.

762
00:48:09 --> 00:48:15
Let me say something more about
that, how to use eigenvalues.

763
00:48:15 --> 00:48:22
Suppose I have this system of
1,000 differential equations.

764
00:48:22 --> 00:48:27
Linear, constant coefficients,
starts from some u(0).

765
00:48:27 --> 00:48:34
 
 

766
00:48:34 --> 00:48:37
How do eigenvalues and
eigenvectors help?

767
00:48:37 --> 00:48:40
Well, first I have to find
them, that's the job.

768
00:48:40 --> 00:48:44
So suppose I find 1,000
eigenvalues and eigenvectors.

769
00:48:44 --> 00:48:50
A times eigenvector number
i is eigenvalue number i

770
00:48:50 --> 00:48:52
times eigenvector number i.
 

771
00:48:52 --> 00:48:58
So these, y_1 to y_1,000,
so y_1 to y_1,000 are

772
00:48:58 --> 00:49:00
the eigenvectors.
 

773
00:49:00 --> 00:49:03
And each one has its own
eigenvalue, lambda_1

774
00:49:03 --> 00:49:04
to lambda_1,000.
 

775
00:49:05 --> 00:49:11
And now if I did that work,
sort of like, in advance,

776
00:49:11 --> 00:49:14
now I come to the
differential equation.

777
00:49:14 --> 00:49:21
How could I use this?
 

778
00:49:21 --> 00:49:27
This is now going to be the
most-- it's three steps to

779
00:49:27 --> 00:49:33
use it, three steps to use
these to get the answer.

780
00:49:33 --> 00:49:37
Ready for step one.
 

781
00:49:37 --> 00:49:44
Step one is break u_0
into eigenvectors.

782
00:49:44 --> 00:49:48
Split, separate, write,
express u(0) as a

783
00:49:48 --> 00:50:02
combination of eigenvectors.
 

784
00:50:02 --> 00:50:05
Now step two.
 

785
00:50:05 --> 00:50:08
What happens to
each eigenvector?

786
00:50:08 --> 00:50:10
So this is where the
differential equation

787
00:50:10 --> 00:50:11
starts from.
 

788
00:50:11 --> 00:50:13
This is the initial condition.
 

789
00:50:13 --> 00:50:19
1,000 components of u at the
start and it's separated into

790
00:50:19 --> 00:50:23
1,000 eigenvector pieces.
 

791
00:50:23 --> 00:50:28
Now step two is watch
each piece separately.

792
00:50:28 --> 00:50:41
So step two will be multiply
say, c_1 by e^(lambda_1*t),

793
00:50:41 --> 00:50:44
by it's growth.
 

794
00:50:44 --> 00:50:47
This is following
eigenvector number one.

795
00:50:47 --> 00:50:50
And in general, I would
multiply every one of the

796
00:50:50 --> 00:50:55
c's by e to those guys.
 

797
00:50:55 --> 00:50:59
So what would I have now?
 

798
00:50:59 --> 00:51:01
This is one piece of the start.
 

799
00:51:01 --> 00:51:05
And that gives me one
piece of the finish.

800
00:51:05 --> 00:51:14
So the finish is, the answer is
to add up the 1,000 pieces.

801
00:51:14 --> 00:51:18
And if you're with me, you see
what those 1,000 pieces are.

802
00:51:18 --> 00:51:23
Here's a piece, some multiple
of the first eigenvector.

803
00:51:23 --> 00:51:27
Now if we only were working
with that piece, we follow it

804
00:51:27 --> 00:51:31
in time by multiplying it by
either the lambda_1 * t,

805
00:51:31 --> 00:51:35
and what do we have
at a later time?

806
00:51:36 --> 00:51:36
c_1*e^(lambda_1*t)y_1.
 

807
00:51:41 --> 00:51:45
This piece has grown into that.
 

808
00:51:45 --> 00:51:48
And other pieces have
grown into other things.

809
00:51:48 --> 00:51:50
And what about the last piece?
 

810
00:51:50 --> 00:51:57
So what is it that
I have to add up?

811
00:51:57 --> 00:51:59
Tell me what to write here.
 

812
00:51:59 --> 00:52:07
c_1,000, however much of
eigenvector 1,000 was in there,

813
00:52:07 --> 00:52:14
and then finally, never written
left-handed before,

814
00:52:14 --> 00:52:20
e to the who?
 

815
00:52:20 --> 00:52:25
Lambda number 1,000, not
1,000 itself, but it's

816
00:52:25 --> 00:52:27
eigenvalue, 1,000t.
 

817
00:52:31 --> 00:52:36
This is just splitting, this
is constantly, constantly

818
00:52:36 --> 00:52:41
the method, the way to use
eigenvalues and eigenvectors.

819
00:52:41 --> 00:52:46
Split the problem into
the pieces that go,

820
00:52:46 --> 00:52:48
that are eigenvectors.
 

821
00:52:48 --> 00:52:53
Watch each piece,
add up the pieces.

822
00:52:53 --> 00:52:56
That's why eigenvectors
are so important.

823
00:52:56 --> 00:52:59
Yeah?
 

824
00:52:59 --> 00:53:02
Yes, right.
 

825
00:53:02 --> 00:53:08
Well, now, very good question.
 

826
00:53:08 --> 00:53:10
Let's see.
 

827
00:53:10 --> 00:53:12
Well, the first thing we
have to know is that we do

828
00:53:12 --> 00:53:14
find 1,000 eigenvectors.
 

829
00:53:14 --> 00:53:19
And so my answer is going to
be for symmetric matrices,

830
00:53:19 --> 00:53:21
everything always works.
 

831
00:53:21 --> 00:53:26
For symmetric matrices, if size
is 1,000, they have 1,000

832
00:53:26 --> 00:53:28
eigenvectors, and next time
we'll have a shot

833
00:53:28 --> 00:53:30
at some of these.
 

834
00:53:30 --> 00:53:33
What some of them are for
these special matrices.

835
00:53:33 --> 00:53:39
So this method always works
if I've got a full family of

836
00:53:39 --> 00:53:42
independent eigenvectors.
 

837
00:53:42 --> 00:53:47
If it's of size 1,000, I need,
you're right, exactly right.

838
00:53:47 --> 00:53:52
To see that this was
the questionable step.

839
00:53:52 --> 00:53:55
If I haven't got 1,000
eigenvectors, I'm not going to

840
00:53:55 --> 00:53:57
be able to take that step.
 

841
00:53:57 --> 00:53:59
And it happens.
 

842
00:53:59 --> 00:54:05
I am sad to report that
some matrices haven't

843
00:54:05 --> 00:54:07
got enough eigenvectors.
 

844
00:54:07 --> 00:54:11
Some matrices, they collapse.
 

845
00:54:11 --> 00:54:15
This always happens
in math, somehow.

846
00:54:15 --> 00:54:20
Two eigenvectors collapse
into one and the matrix is

847
00:54:20 --> 00:54:23
defective, like it's a loser.
 

848
00:54:23 --> 00:54:28
So now you have to, of
course, the equation

849
00:54:28 --> 00:54:31
still has a solution.
 

850
00:54:31 --> 00:54:37
So there has to be something
there, but the pure eigenvector

851
00:54:37 --> 00:54:40
method is not going to make
it on those special matrices.

852
00:54:40 --> 00:54:44
I could write down one
but why should we give

853
00:54:44 --> 00:54:46
space to a loser?
 

854
00:54:46 --> 00:54:51
But what happens in that case?
 

855
00:54:51 --> 00:54:55
You might remember from
differential equations when two

856
00:54:55 --> 00:54:59
of these roots, these are like
roots, these lambdas are like

857
00:54:59 --> 00:55:04
roots that you found in solving
a differential equation.

858
00:55:04 --> 00:55:09
When two of them come together,
that's when the danger is.

859
00:55:09 --> 00:55:11
When I have a double
eigenvalue, then there's a

860
00:55:11 --> 00:55:15
high risk that I've only
got one eigenvector.

861
00:55:15 --> 00:55:21
And I'll just put in this
little thing what the other, so

862
00:55:21 --> 00:55:23
the e^(lambda_1*t) is fine.
 

863
00:55:23 --> 00:55:29
But if that y_1 is like, if the
lambda_1's in there twice,

864
00:55:29 --> 00:55:30
I need something new.
 

865
00:55:30 --> 00:55:34
And the new thing turns out
to be t*e^(lambda* t).

866
00:55:34 --> 00:55:38
 
 

867
00:55:38 --> 00:55:40
I don't know if
anybody remembers.

868
00:55:40 --> 00:55:44
This was probably hammered back
in differential equations that

869
00:55:44 --> 00:55:50
if you had repeated something
or other then this, you didn't

870
00:55:50 --> 00:55:53
get pure e^(lambda*t)'s, you
got also a t*e^(lambda*t).

871
00:55:54 --> 00:55:56
Anyway that's the answer.
 

872
00:55:56 --> 00:55:59
That if we're short
eigenvectors, and it can

873
00:55:59 --> 00:56:02
happen, but it won't
for our good matrices.

874
00:56:02 --> 00:56:07
Ok, so Monday I've
got lots to do.

875
00:56:07 --> 00:56:11
Special eigenvalues and vectors
and then positive definite.