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PROFESSOR STRANG: Finally
we get to positive
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definite matrices.
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I've used the word and now
it's time to pin it down.
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And so this would be my thank
you for staying with it while
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we do this important
preliminary stuff
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about linear algebra.
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So starting the next lecture
we'll really make a big start
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on engineering applications.
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But these matrices are going
to be the key to everything.
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And I'll call these matrices K
and positive definite, I will
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00:00:58 --> 00:01:03
only use that word about
a symmetric matrix.
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So the matrix is already
symmetric and that means it
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has real eigenvalues and many
other important properties,
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orthogonal eigenvectors.
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And now we're asking for more.
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And it's that extra bit
that is terrific in all
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kinds of applications.
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00:01:28 --> 00:01:31
So if I can do this bit
of linear algebra.
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So what's coming then, my
review session this afternoon
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00:01:34 --> 00:01:41
at four, I'm very happy that
we've got, I think, the
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best MATLAB problem ever
invented in 18.085 anyway.
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So that should get onto the
website probably by tomorrow.
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00:01:51 --> 00:01:55
And Peter Buchak is like
the MATLAB person.
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So his review sessions
are Friday at noon.
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And I just saw him and
suggested Friday at
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noon he might as well
just stay in here.
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00:02:06 --> 00:02:10
And knowing that that
isn't maybe a good
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hour for everybody.
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00:02:11 --> 00:02:16
So you could see him also
outside of that hour.
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00:02:16 --> 00:02:20
But that's the hour he will
be ready for all kinds of
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00:02:20 --> 00:02:23
questions about MATLAB
or about the homeworks.
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00:02:23 --> 00:02:31
Actually you'll be probably
thinking more also about the
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homework questions
on this topic.
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00:02:35 --> 00:02:40
Ready for positive definite?
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You said yes, right?
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00:02:43 --> 00:02:50
And you have a hint
about these things.
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00:02:50 --> 00:02:55
So we have a symmetric matrix
and the beauty is that it
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00:02:55 --> 00:02:58
brings together all
of linear algebra.
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Including elimination,
that's when we see pivots.
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00:03:01 --> 00:03:04
Including determinants
which are closely
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related to the pivots.
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00:03:05 --> 00:03:08
And what do I mean
by upper left?
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I mean that if I have a three
by three symmetric matrix and I
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00:03:13 --> 00:03:17
want to test it for positive
definite, and I guess actually
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this would be the easiest test
if I had a tiny matrix, three
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by three, and I had numbers
then this would be a good test.
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The determinants, by upper
left determinants I mean
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that one by one determinant.
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So just that first number
has to be positive.
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00:03:36 --> 00:03:39
Then the two by two
determinant, that times
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that minus that times
that has to be positive.
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Oh I've already
been saying that.
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Let me just put
in some letters.
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So a has to be positive.
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This is symmetric, so
a times c has to be
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bigger than b squared.
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So that will tell us.
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And then for two
by two we finish.
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For three by three we would
also require the three by three
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determinant to be positive.
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But already here you're
seeing one point about a
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positive definite matrix.
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Its diagonal will
have to be positive.
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And somehow its diagonal has to
be not just above zero, but
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somehow it has to
defeat b squared.
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So the diagonal has to be
somehow more positive than
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whatever negative stuff might
come from off the diagonal.
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That's why I would
need a*c > b squared.
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So both of those will
be positive and their
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product has to be bigger
than the other guy.
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And then finally, a third
test is that all the
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eigenvalues are positive.
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00:04:53 --> 00:04:56
And of course if I give you a
three by three matrix, that's
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00:04:56 --> 00:04:59
probably not the easiest
test since you'd have to
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00:04:59 --> 00:05:00
find the eigenvalues.
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00:05:00 --> 00:05:05
Much easier to find the
determinants or the pivots.
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00:05:05 --> 00:05:09
Actually, just while I'm
at it, so the first pivot
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of course is a itself.
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No difficulty there.
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The second pivot turns
out to be the ratio of
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00:05:18 --> 00:05:22
a*c - b squared to a.
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So the connection between
pivots and determinants
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00:05:25 --> 00:05:28
is just really close.
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00:05:28 --> 00:05:30
Pivots are ratios
of determinants if
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00:05:30 --> 00:05:31
you work it out.
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00:05:31 --> 00:05:36
The second pivot, maybe I would
call that d_2, is the ratio
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00:05:36 --> 00:05:40
of a*c - b squared over a.
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00:05:40 --> 00:05:41
In other words it's
(c - b squared)/a.
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00:05:41 --> 00:05:48
98
00:05:48 --> 00:05:51
Determinants are positive
and vice versa.
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00:05:51 --> 00:05:54
Then it's fantastic that
the eigenvalues come
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00:05:54 --> 00:05:56
into the picture.
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00:05:56 --> 00:06:00
So those are three ways, three
important properties of a
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positive definite matrix.
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00:06:02 --> 00:06:08
But I'd like to make the
definition something different.
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00:06:08 --> 00:06:11
Now I'm coming to the meaning.
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If I think of those as
the tests, that's done.
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00:06:14 --> 00:06:24
Now the meaning of
positive definite.
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The meaning of positive
definite and the applications
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are closely related to
looking for a minimum.
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00:06:32 --> 00:06:39
And so what I'm going to bring
in here, so it's symmetric.
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00:06:39 --> 00:06:47
Now for a symmetric matrix I
want to introduce the energy.
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00:06:47 --> 00:06:51
So this is the reason why it
has so many applications and
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00:06:51 --> 00:06:56
such important physical meaning
is that what I'm about to
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introduce, which is a function
of x, and here it is, it's x
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00:07:02 --> 00:07:11
transpose times A, not A, I'm
sticking with K for
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00:07:11 --> 00:07:16
my matrix, times x.
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I think of that as some f(x).
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And let's just see what it
would be if the matrix was
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00:07:24 --> 00:07:29
two by two, [a, b; b, c].
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Suppose that's my matrix.
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We want to get a handle on
what, this is the first time
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00:07:36 --> 00:07:42
I've ever written something
that has x's times x's.
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00:07:42 --> 00:07:45
So it's going to be quadratic.
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00:07:45 --> 00:07:48
They're going to
be x's times x's.
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00:07:48 --> 00:07:52
And x is a general vector
of the right size so it's
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got components x_1, x_2.
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00:07:55 --> 00:07:57
And there it's transpose,
so it's a row.
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00:07:57 --> 00:08:01
And now I put in
the [a, b; b, c].
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And then I put in x again.
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This is going to give me
a very nice, simple,
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00:08:08 --> 00:08:11
important expression.
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Depending on x_1 and x_2.
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Now what is, can we do
that multiplication?
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Maybe above I'll do the
multiplication of this
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pair and then I have the
other guy to bring in.
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00:08:28 --> 00:08:30
So here, that would
be ax_1+bx_2.
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00:08:30 --> 00:08:35
137
00:08:35 --> 00:08:36
And this would be bx_1+cx_2.
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00:08:36 --> 00:08:39
139
00:08:39 --> 00:08:44
So that's the first,
that's this times this.
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00:08:44 --> 00:08:45
What am I going to get?
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00:08:45 --> 00:08:48
What shape, what size is
this result going to be?
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00:08:48 --> 00:08:56
This K is n by n. x is a column
vector. n by one. x transpose,
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00:08:56 --> 00:08:58
what's the shape
of x transpose?
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00:08:58 --> 00:09:00
One by n?
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00:09:00 --> 00:09:02
So what's the total result?
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00:09:02 --> 00:09:03
One by one.
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00:09:03 --> 00:09:04
Just a number.
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00:09:04 --> 00:09:05
Just a function.
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00:09:05 --> 00:09:06
It's a number.
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00:09:06 --> 00:09:11
But it depends on the x's
and the matrix inside.
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Can we do it now?
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00:09:12 --> 00:09:16
So I've got this to
multiply by this.
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Do you see an x_1
squared showing up?
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00:09:19 --> 00:09:21
Yes, from there times there.
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00:09:21 --> 00:09:24
And what's it multiplied by?
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00:09:24 --> 00:09:26
The a.
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00:09:26 --> 00:09:30
The first term is this times
the ax_1 is a(x_1 squared).
158
00:09:30 --> 00:09:33
So that's our first quadratic.
159
00:09:33 --> 00:09:35
Now there'd be an x_1, x_2.
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00:09:36 --> 00:09:39
Let me leave that for a minute
and find the x_2 squared
161
00:09:39 --> 00:09:41
because it's easy.
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00:09:41 --> 00:09:43
So where am I going
to get x_2 squared?
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00:09:43 --> 00:09:47
I'm going to get that from
x_2, second guy here
164
00:09:47 --> 00:09:49
times second guy here.
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00:09:49 --> 00:09:54
There's a c(x_2 squared).
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00:09:54 --> 00:09:58
So you're seeing already
where the diagonal shows up.
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00:09:58 --> 00:10:02
The diagonal a, c, whatever
is multiplying the
168
00:10:02 --> 00:10:04
perfect squares.
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00:10:04 --> 00:10:08
And it'll be the off-diagonal
that multiplies the x_1, x_2.
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00:10:08 --> 00:10:11
We might call those
the crossterms.
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00:10:11 --> 00:10:13
And what do we get
for that then?
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00:10:13 --> 00:10:16
We have x_1 times this guy.
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00:10:16 --> 00:10:20
So that's a crossterm.
bx_1*x_2, right?
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00:10:20 --> 00:10:24
And here's another one coming
from x_2 times this guy.
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00:10:24 --> 00:10:27
And what's that one?
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00:10:27 --> 00:10:29
It's also bx_1*x_2.
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00:10:30 --> 00:10:34
So x_1, multiply that, x_2
multiply that, and so what
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00:10:34 --> 00:10:37
do we have for this
crossterm here?
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00:10:37 --> 00:10:38
Two of them.
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00:10:39 --> 00:10:40
2bx_1*x_2.
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00:10:43 --> 00:10:48
In other words, that b
and that b came together
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00:10:48 --> 00:10:49
in the 2bx_1*x_2.
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00:10:50 --> 00:10:56
So here's my energy.
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00:10:56 --> 00:10:58
Can I just loosely
call it energy?
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00:10:58 --> 00:11:02
And then as we get to
applications we'll see why.
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00:11:02 --> 00:11:06
So I'm interested in
that because it has
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00:11:06 --> 00:11:12
important meaning.
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00:11:12 --> 00:11:15
Well, so now I'm ready
to define positive
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00:11:15 --> 00:11:16
definite matrices.
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00:11:16 --> 00:11:19
So I'll call that number four.
191
00:11:19 --> 00:11:23
But I'm going to
give it a big star.
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00:11:23 --> 00:11:29
Even more because it's
the sort of key.
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00:11:29 --> 00:11:34
So the test will be, you can
probably guess it, I look at
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00:11:34 --> 00:11:40
this expression, that
x transpose Ax.
195
00:11:41 --> 00:11:46
And if it's a positive definite
matrix and this represents
196
00:11:46 --> 00:11:50
energy, the key will be that
this should be positive.
197
00:11:50 --> 00:11:57
This one should be
positive for all x's.
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00:11:57 --> 00:11:59
Well, with one
exception, of course.
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00:11:59 --> 00:12:06
All x's except, which vector
is it? x=0 would just
200
00:12:06 --> 00:12:10
give me-- See, I put K.
201
00:12:10 --> 00:12:17
My default for a matrix,
but should be, it's K.
202
00:12:17 --> 00:12:22
Except x=0, except
the zero vector.
203
00:12:22 --> 00:12:23
Of course.
204
00:12:23 --> 00:12:29
If x_1 and x_2 are both zero.
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00:12:29 --> 00:12:34
Now that looks a little maybe
less straightforward, I would
206
00:12:34 --> 00:12:38
say, because it's a statement
about this is true
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00:12:38 --> 00:12:40
for all x_1 and x_2.
208
00:12:41 --> 00:12:44
And we better do some
examples and draw a picture.
209
00:12:44 --> 00:12:51
Let me draw a
picture right away.
210
00:12:51 --> 00:12:54
So here's x_1 direction.
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00:12:54 --> 00:12:56
Here's x_2 direction.
212
00:12:56 --> 00:13:05
And here is the x transpose
Ax, my function.
213
00:13:05 --> 00:13:09
So this depends on
two variables.
214
00:13:09 --> 00:13:13
So it's going to be a sort
of a surface if I draw it.
215
00:13:13 --> 00:13:16
Now, what point do
we absolutely know?
216
00:13:16 --> 00:13:21
And I put A again.
217
00:13:21 --> 00:13:29
I am so sorry.
218
00:13:29 --> 00:13:30
Well, we know one point.
219
00:13:30 --> 00:13:34
It's there whatever
that matrix might be.
220
00:13:34 --> 00:13:35
It's there.
221
00:13:35 --> 00:13:37
Zero, right?
222
00:13:37 --> 00:13:40
You just told me that if
both x's are zero then we
223
00:13:40 --> 00:13:42
automatically get zero.
224
00:13:42 --> 00:13:47
Now what do you think the shape
of this curve, the shape of
225
00:13:47 --> 00:13:51
this graph is going
to look like?
226
00:13:51 --> 00:13:54
The point is, if we're
positive definite now.
227
00:13:54 --> 00:13:58
So I'm drawing the picture
for positive definite.
228
00:13:58 --> 00:14:04
So my definition is that
the energy goes up.
229
00:14:04 --> 00:14:06
It's positive, right?
230
00:14:06 --> 00:14:10
When I leave, when I move away
from that point I go upwards.
231
00:14:10 --> 00:14:13
That point will be a minimum.
232
00:14:13 --> 00:14:17
Let me just draw it roughly.
233
00:14:17 --> 00:14:23
So it sort of goes
up like this.
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00:14:23 --> 00:14:30
These cheap 2-D boards and
I've got a three-dimensional
235
00:14:30 --> 00:14:34
picture here.
236
00:14:34 --> 00:14:35
But you see it somehow?
237
00:14:35 --> 00:14:40
What word or what's
your visualization?
238
00:14:40 --> 00:14:43
It has a minimum there.
239
00:14:43 --> 00:14:46
That's why minimization, which
was like, the core problem
240
00:14:46 --> 00:14:49
in calculus, is here now.
241
00:14:49 --> 00:14:54
But for functions of
two x's or n x's.
242
00:14:54 --> 00:14:58
We're up the dimension
over the basic minimum
243
00:14:58 --> 00:15:03
problem of calculus.
244
00:15:03 --> 00:15:06
It's sort of like a parabola
It's cross-sections cutting
245
00:15:06 --> 00:15:09
down through the thing would
be just parabolas because
246
00:15:09 --> 00:15:10
of the x squared.
247
00:15:10 --> 00:15:12
I'm going to call this a bowl.
248
00:15:12 --> 00:15:16
It's a short word.
249
00:15:16 --> 00:15:16
Do you see it?
250
00:15:16 --> 00:15:18
It opens up.
251
00:15:18 --> 00:15:20
That's the key point,
that it opens upward.
252
00:15:20 --> 00:15:22
And let's do some examples.
253
00:15:22 --> 00:15:26
Tell me some positive definite.
254
00:15:26 --> 00:15:30
So positive definite and then
let me here put some not
255
00:15:30 --> 00:15:35
positive definite cases.
256
00:15:35 --> 00:15:38
Tell me a matrix.
257
00:15:38 --> 00:15:42
Well, what's the easiest, first
matrix that occurs to you as
258
00:15:42 --> 00:15:44
a positive definite matrix?
259
00:15:44 --> 00:15:49
The identity.
260
00:15:49 --> 00:15:52
That passes all our tests,
its eigenvalues are one,
261
00:15:52 --> 00:15:54
its pivots are one, the
determinants are one.
262
00:15:54 --> 00:16:00
And the function is x_1
squared plus x_2 squared
263
00:16:00 --> 00:16:05
with no b in it.
264
00:16:05 --> 00:16:08
It's just a perfect bowl,
perfectly symmetric, the
265
00:16:08 --> 00:16:12
way it would come off
a potter's wheel.
266
00:16:12 --> 00:16:16
Now let me take one that's
maybe not so, let me
267
00:16:16 --> 00:16:18
put a nine there.
268
00:16:18 --> 00:16:20
So I'm off to a
reasonable start.
269
00:16:20 --> 00:16:22
I have an x_1 squared
and a nine x_2 squared.
270
00:16:24 --> 00:16:27
And now I want to ask you, what
could I put in there that would
271
00:16:27 --> 00:16:30
leave it positive definite?
272
00:16:30 --> 00:16:33
Well, give me a couple
of possibilities.
273
00:16:33 --> 00:16:37
What's a nice, not too big
now, that's the thing.
274
00:16:37 --> 00:16:38
Two.
275
00:16:38 --> 00:16:39
Two would be fine.
276
00:16:39 --> 00:16:42
So if I had a two there and a
two there I would have a
277
00:16:42 --> 00:16:47
4x_1*x_2 and it would, like,
this, instead of being a
278
00:16:47 --> 00:16:55
circle, which it was for the
identity, the plane there would
279
00:16:55 --> 00:16:59
cut out a ellipse instead.
280
00:16:59 --> 00:17:02
But it would be a good ellipse.
281
00:17:02 --> 00:17:06
Because we're doing squares,
we're really, the Greeks
282
00:17:06 --> 00:17:11
understood these second degree
things and they would have
283
00:17:11 --> 00:17:18
known this would have
been an ellipse.
284
00:17:18 --> 00:17:23
How high can I go with that two
or where do I have to stop?
285
00:17:23 --> 00:17:27
Where would I have to, if I
wanted to change the two, let
286
00:17:27 --> 00:17:33
me just focus on that one,
suppose I wanted to change it.
287
00:17:33 --> 00:17:36
First of all, give me
one that's, how about
288
00:17:36 --> 00:17:38
the borderline.
289
00:17:38 --> 00:17:40
Three would be the borderline.
290
00:17:40 --> 00:17:41
Why's that?
291
00:17:41 --> 00:17:48
Because at three we have nine
minus nine for the determinant.
292
00:17:48 --> 00:17:51
So the determinant is zero.
293
00:17:51 --> 00:17:53
Of course it passed
the first test.
294
00:17:53 --> 00:17:54
One by one was ok.
295
00:17:54 --> 00:18:03
But two by two was not,
was at the borderline.
296
00:18:03 --> 00:18:07
What else should I think?
297
00:18:07 --> 00:18:11
Oh, that's a very
interesting case.
298
00:18:11 --> 00:18:13
The borderline.
299
00:18:13 --> 00:18:15
You know, it almost makes it.
300
00:18:15 --> 00:18:21
But can you tell me the
eigenvalues of that matrix?
301
00:18:21 --> 00:18:25
Don't do any
quadratic equations.
302
00:18:25 --> 00:18:28
How do I know, what's one
eigenvalue of a matrix?
303
00:18:28 --> 00:18:30
You made it singular, right?
304
00:18:30 --> 00:18:31
You made that matrix singular.
305
00:18:31 --> 00:18:32
Determinant zero.
306
00:18:32 --> 00:18:36
So one of its
eigenvalues is zero.
307
00:18:36 --> 00:18:40
And the other one is visible
by looking at the trace.
308
00:18:40 --> 00:18:44
I just quickly mentioned that
if I add the diagonal, I
309
00:18:44 --> 00:18:47
get the same answer as if I
add the two eigenvalues.
310
00:18:47 --> 00:18:51
So that other eigenvalue
must be ten.
311
00:18:51 --> 00:18:55
And this is entirely typical,
that ten and zero, the extreme
312
00:18:55 --> 00:19:01
eigenvalues, lambda_max and
lambda_min, are bigger than,
313
00:19:01 --> 00:19:04
these diagonal guys are inside.
314
00:19:04 --> 00:19:09
They're inside, between zero
and ten and it's these terms
315
00:19:09 --> 00:19:13
that enter somehow and gave
us an eigenvalue of ten
316
00:19:13 --> 00:19:15
and an eigenvalue of zero.
317
00:19:15 --> 00:19:20
I guess I'm tempted to
try to draw that figure.
318
00:19:20 --> 00:19:25
Just to get a feeling of
what's with that one.
319
00:19:25 --> 00:19:30
It always helps to get
the borderline case.
320
00:19:30 --> 00:19:32
So what's with this one?
321
00:19:32 --> 00:19:35
Let me see what my
quadratic would be.
322
00:19:35 --> 00:19:37
Can I just change it up here?
323
00:19:37 --> 00:19:38
Rather than rewriting it.
324
00:19:38 --> 00:19:42
So I'm going to, I'll
put it up here.
325
00:19:42 --> 00:19:46
So I have to change
that four to what?
326
00:19:46 --> 00:19:48
Now that I'm looking
at this matrix.
327
00:19:48 --> 00:19:51
That four is now a six.
328
00:19:51 --> 00:19:53
Six.
329
00:19:53 --> 00:19:56
This is my guy for this one.
330
00:19:56 --> 00:19:58
Which is not positive definite.
331
00:19:58 --> 00:20:00
Let me tell you right away
the word that I would
332
00:20:00 --> 00:20:02
use for this one.
333
00:20:02 --> 00:20:06
I would call it positive
semi-definite because it's
334
00:20:06 --> 00:20:09
almost there, but not quite.
335
00:20:09 --> 00:20:15
So semi-definite allows the
matrix to be singular.
336
00:20:15 --> 00:20:19
So semi-definite, maybe
I'll do it in green what
337
00:20:19 --> 00:20:22
semi-definite would be.
338
00:20:22 --> 00:20:31
Semi-def would be eigenvalues
greater than or equal zero.
339
00:20:31 --> 00:20:35
Determinants greater
than or equal zero.
340
00:20:35 --> 00:20:39
Pivots greater than zero
if they're there or then
341
00:20:39 --> 00:20:41
we run out of pivots.
342
00:20:41 --> 00:20:44
You could say greater than
or equal to zero then.
343
00:20:44 --> 00:20:48
And energy, greater
than or equal to zero
344
00:20:48 --> 00:20:53
for semi-definite.
345
00:20:53 --> 00:20:58
And when would the energy, what
x's, what would be the like,
346
00:20:58 --> 00:21:02
you could say the ground states
or something, what x's, so
347
00:21:02 --> 00:21:06
greater than or equal to zero,
emphasize that possibility
348
00:21:06 --> 00:21:10
of equal in the
semi-definite case.
349
00:21:10 --> 00:21:17
Suppose I have a semi-definite
matrix, yeah, I've got one.
350
00:21:17 --> 00:21:19
But it's singular.
351
00:21:19 --> 00:21:26
So that means a singular matrix
takes some vector x to zero.
352
00:21:26 --> 00:21:27
Right?
353
00:21:27 --> 00:21:30
If my matrix is actually
singular, then there'll be
354
00:21:30 --> 00:21:32
an x where Kx is zero.
355
00:21:32 --> 00:21:35
And then, of course,
multiplying by x transpose,
356
00:21:35 --> 00:21:36
I'm still at zero.
357
00:21:36 --> 00:21:41
So the x's, the zero energy
guys, this is straightforward,
358
00:21:41 --> 00:21:49
the zero energy guys, the ones
where x transpose Kx is zero,
359
00:21:49 --> 00:21:56
will happen when Kx is zero.
360
00:21:56 --> 00:22:03
If Kx is zero, and we'll
see it in that example.
361
00:22:03 --> 00:22:05
Let's see it in that example.
362
00:22:05 --> 00:22:12
What's the x for which,
I could say in the null
363
00:22:12 --> 00:22:18
space, what's the vector
x that that matrix kills?
364
00:22:18 --> 00:22:21
365
00:22:21 --> 00:22:23
, right?
366
00:22:23 --> 00:22:24
The vector .
367
00:22:24 --> 00:22:28
368
00:22:28 --> 00:22:30
gives me .
369
00:22:30 --> 00:22:33
That's the vector that,
so I get 3-3, the
370
00:22:33 --> 00:22:36
zero, 9-9, the zero.
371
00:22:36 --> 00:22:42
So I believe that this
thing will be-- Is it
372
00:22:42 --> 00:22:44
zero at three, minus one?
373
00:22:44 --> 00:22:47
I think that it
has to be, right?
374
00:22:47 --> 00:22:52
If I take x_1 to be three and
x_2 to be minus one, I think
375
00:22:52 --> 00:22:54
I've got zero energy here.
376
00:22:54 --> 00:22:58
Do I? x_1 squared will be
at the nine and nine x_2
377
00:22:58 --> 00:23:01
squared will be nine more.
378
00:23:01 --> 00:23:02
And what will be this 6x_1*x_2?
379
00:23:04 --> 00:23:09
What will that come out
for this x_1 and x_2?
380
00:23:09 --> 00:23:10
Minus 18.
381
00:23:10 --> 00:23:11
Had to, right?
382
00:23:11 --> 00:23:13
So I'd get nine from
there, nine from
383
00:23:13 --> 00:23:15
there, minus 18, zero.
384
00:23:15 --> 00:23:18
So the graph for this
positive semi-definite
385
00:23:18 --> 00:23:21
will look a bit like this.
386
00:23:21 --> 00:23:26
There'll be a direction in
which it doesn't climb.
387
00:23:26 --> 00:23:29
It doesn't go below
the base, right?
388
00:23:29 --> 00:23:31
It's never negative.
389
00:23:31 --> 00:23:33
This is now the
semi-definite picture.
390
00:23:33 --> 00:23:36
But it can run along the base.
391
00:23:36 --> 00:23:40
And it will for the vector
x_1=3, x_2=-1, I don't know
392
00:23:40 --> 00:23:45
where that is, one, two, three,
and then maybe minus one.
393
00:23:45 --> 00:23:51
Along some line here the
graph doesn't go up.
394
00:23:51 --> 00:23:56
It's sitting, can you imagine
that sitting in the base?
395
00:23:56 --> 00:24:05
I'm not Rembrandt here, but in
the other direction it goes up.
396
00:24:05 --> 00:24:08
Oh, the hell with that one.
397
00:24:08 --> 00:24:10
Do you see, sort of?
398
00:24:10 --> 00:24:14
It's like a trough,
would you say?
399
00:24:14 --> 00:24:16
I mean, it's like a,
you know, a bit of a
400
00:24:16 --> 00:24:19
drainpipe or something.
401
00:24:19 --> 00:24:27
It's running along the ground,
along this direction and
402
00:24:27 --> 00:24:30
in the other directions
it does go up.
403
00:24:30 --> 00:24:35
So it's shaped like this
with the base not climbing.
404
00:24:35 --> 00:24:39
Whereas here, there's
no bad direction.
405
00:24:39 --> 00:24:40
Climbs every way you go.
406
00:24:40 --> 00:24:45
So that's positive definite and
that's positive semi-definite.
407
00:24:45 --> 00:24:50
Well suppose I push
it a little further.
408
00:24:50 --> 00:24:56
Let me make a place here for
a matrix that isn't even
409
00:24:56 --> 00:25:01
positive semi-definite.
410
00:25:01 --> 00:25:05
Now it's just going to
go down somewhere.
411
00:25:05 --> 00:25:07
I'll start again with one
and nine and tell me
412
00:25:07 --> 00:25:09
what to put in now.
413
00:25:09 --> 00:25:13
So this is going to be a
case where the off-diagonal
414
00:25:13 --> 00:25:15
is too big, it wins.
415
00:25:15 --> 00:25:18
And prevents positive definite.
416
00:25:18 --> 00:25:21
So what number would
you like here?
417
00:25:21 --> 00:25:22
Five?
418
00:25:22 --> 00:25:27
Five is certainly plenty.
419
00:25:27 --> 00:25:30
So now I have [1, 5; 5, 9].
420
00:25:30 --> 00:25:37
Let me take a little space on
a board just to show you.
421
00:25:37 --> 00:25:42
Sorry about that.
422
00:25:42 --> 00:25:46
So I'm going to do the [1, 5;
5, 9] just because they're all
423
00:25:46 --> 00:25:48
important, but then
we're coming back to
424
00:25:48 --> 00:25:49
positive definite.
425
00:25:49 --> 00:25:59
So if it's [1, 5; 5, 9] and I
do that usual x, x transpose Kx
426
00:25:59 --> 00:26:03
and I do the multiplication
out, I see the one x_1 squared
427
00:26:03 --> 00:26:06
and I see the nine
x_2 squareds.
428
00:26:06 --> 00:26:11
And how many
x_1*x_2's do I see?
429
00:26:11 --> 00:26:15
Five from there, five
from there, ten.
430
00:26:15 --> 00:26:20
And I believe that
can be negative.
431
00:26:20 --> 00:26:24
The fact of having all nice
plus signs is not going to help
432
00:26:24 --> 00:26:28
it because we can choose, as we
already did, x_1 to be like
433
00:26:28 --> 00:26:31
a negative number and
x_2 to be a positive.
434
00:26:31 --> 00:26:35
And we can get this guy to be
negative and make it, in this
435
00:26:35 --> 00:26:41
case we can make it defeat
these positive parts.
436
00:26:41 --> 00:26:43
What choice would do it?
437
00:26:43 --> 00:26:49
Let me take x_1 to be minus one
and tell me an x_2 that's good
438
00:26:49 --> 00:26:55
enough to show that this thing
is not positive definite or
439
00:26:55 --> 00:26:58
even semi-definite,
it goes downhill.
440
00:26:58 --> 00:26:59
Take x_2 equal?
441
00:26:59 --> 00:27:04
What do you say?
442
00:27:04 --> 00:27:05
1/2?
443
00:27:05 --> 00:27:08
Yeah, I don't want too big an
x_2 because if I have too big
444
00:27:08 --> 00:27:10
an x_2, then this'll
be important.
445
00:27:10 --> 00:27:14
Does 1/2 do it?
446
00:27:14 --> 00:27:18
So I've got 1/4, that's
positive, but not very.
447
00:27:18 --> 00:27:23
9/4, so I'm up to 10/4,
but this guy is what?
448
00:27:23 --> 00:27:26
Ten and the minus
is minus five.
449
00:27:26 --> 00:27:27
Yeah.
450
00:27:27 --> 00:27:31
So that absolutely goes,
at this one I come
451
00:27:31 --> 00:27:35
out less than zero.
452
00:27:35 --> 00:27:37
And I might as well complete.
453
00:27:37 --> 00:27:43
So this is the case where I
would call it indefinite.
454
00:27:43 --> 00:27:45
Indefinite.
455
00:27:45 --> 00:27:50
It goes up like if x_2 is
zero, then it's just got
456
00:27:50 --> 00:27:52
x_1 squared, that's up.
457
00:27:52 --> 00:27:55
If x_1 is zero, it's only
got x_2 squared, that's up.
458
00:27:55 --> 00:27:58
But there are other directions
where it goes downhill.
459
00:27:58 --> 00:28:01
So it goes either up, it
goes both up in some
460
00:28:01 --> 00:28:03
ways and down in others.
461
00:28:03 --> 00:28:08
And what kind of a graph, what
kind of a surface would I now
462
00:28:08 --> 00:28:15
have for x transpose for this x
transpose, this indefinite guy?
463
00:28:15 --> 00:28:26
So up in some ways
and down in others.
464
00:28:26 --> 00:28:35
This gets really hard to draw.
465
00:28:35 --> 00:28:40
I believe that if you ride
horses you have an edge
466
00:28:40 --> 00:28:43
on visualizing this.
467
00:28:43 --> 00:28:45
So it's called, what kind
of a point's it called?
468
00:28:45 --> 00:28:50
Saddle point, it's
called a saddle point.
469
00:28:50 --> 00:28:53
So what's a saddle point?
470
00:28:53 --> 00:28:56
That's not bad, right?
471
00:28:56 --> 00:28:58
So this is a direction
where it went up.
472
00:28:58 --> 00:29:02
This is a direction
where it went down.
473
00:29:02 --> 00:29:09
And so it sort of
fills in somehow.
474
00:29:09 --> 00:29:18
Or maybe, if you don't, I
mean, who rides horses now?
475
00:29:18 --> 00:29:25
Actually maybe something we do
do is drive over mountains.
476
00:29:25 --> 00:29:35
So the path, if the road is
sort of well-chosen, the road
477
00:29:35 --> 00:29:42
will go, it'll look for the,
this would be-- Yeah,
478
00:29:42 --> 00:29:43
here's our road.
479
00:29:43 --> 00:29:46
We would do as little
climbing as possible.
480
00:29:46 --> 00:29:48
The mountain would go
like this, sort of.
481
00:29:48 --> 00:29:53
So this would be like, the
bottom part looking along
482
00:29:53 --> 00:29:55
the peaks of the mountains.
483
00:29:55 --> 00:29:59
But it's the top part looking
along the driving direction.
484
00:29:59 --> 00:30:05
So driving, it's a maximum,
but in the mountain range
485
00:30:05 --> 00:30:06
direction it's a minimum.
486
00:30:06 --> 00:30:10
So it's a saddle point.
487
00:30:10 --> 00:30:14
So that's what you get from
a typical symmetric matrix.
488
00:30:14 --> 00:30:19
And if it was minus five it
would still be the same saddle
489
00:30:19 --> 00:30:25
point, would still be 9-25, it
would still be negative
490
00:30:25 --> 00:30:27
and a saddle.
491
00:30:27 --> 00:30:30
Positive guys are our thing.
492
00:30:30 --> 00:30:32
Alright.
493
00:30:32 --> 00:30:36
So now back to
positive definite.
494
00:30:36 --> 00:30:40
With these four tests
and then the discussion
495
00:30:40 --> 00:30:45
of semi-definite.
496
00:30:45 --> 00:30:49
Very key, that energy.
497
00:30:49 --> 00:30:51
Let me just look
ahead a moment.
498
00:30:51 --> 00:30:58
Most physical problems, many,
many physical problems,
499
00:30:58 --> 00:31:00
you have an option.
500
00:31:00 --> 00:31:04
Either you solve some
equations, either you find
501
00:31:04 --> 00:31:10
the solution from our
equations, Ku=f, typically.
502
00:31:10 --> 00:31:12
Matrix equation or
differential equation.
503
00:31:12 --> 00:31:20
Or there's another option of
minimizing some function.
504
00:31:20 --> 00:31:23
Some energy.
505
00:31:23 --> 00:31:25
And it gives the
same equations.
506
00:31:25 --> 00:31:32
So this minimizing energy
will be a second way to
507
00:31:32 --> 00:31:36
describe the applications.
508
00:31:36 --> 00:31:40
Now can I get a number five?
509
00:31:40 --> 00:31:44
There's an important number
five and then you know
510
00:31:44 --> 00:31:48
really all you need to know
about symmetric matrices.
511
00:31:48 --> 00:31:51
This gives me, about positive
definite matrices, this
512
00:31:51 --> 00:32:01
gives me a chance to recap.
513
00:32:01 --> 00:32:06
So I'm going to put
down a number five.
514
00:32:06 --> 00:32:16
Because this is where
the matrices come from.
515
00:32:16 --> 00:32:17
Really important.
516
00:32:17 --> 00:32:20
And it's where they'll come
from in all these applications
517
00:32:20 --> 00:32:23
that chapter two is going to be
all about, that we're
518
00:32:23 --> 00:32:25
going to start.
519
00:32:25 --> 00:32:29
So they come, these positive
definite matrices, so this is
520
00:32:29 --> 00:32:36
another way to, it's a test for
positive definite matrices
521
00:32:36 --> 00:32:40
and it's, actually, it's
where they come from.
522
00:32:40 --> 00:32:44
So here's a positive
definite matrix.
523
00:32:44 --> 00:32:52
They come from A transpose A.
524
00:32:52 --> 00:32:56
A fundamental message is that
if I have just an average
525
00:32:56 --> 00:33:00
matrix, possibly rectangular,
could be a square but not
526
00:33:00 --> 00:33:07
symmetric, then sooner or
later, in fact usually sooner,
527
00:33:07 --> 00:33:10
you end up looking
at A transpose A.
528
00:33:10 --> 00:33:11
We've seen that already.
529
00:33:11 --> 00:33:15
And we already know that A
transpose A is square, we
530
00:33:15 --> 00:33:18
already know it's symmetric and
now we're going to know that
531
00:33:18 --> 00:33:20
it's positive definite.
532
00:33:20 --> 00:33:25
So matrices like A transpose
A are positive definite or
533
00:33:25 --> 00:33:28
possibly semi-definite.
534
00:33:28 --> 00:33:29
There's that possibility.
535
00:33:29 --> 00:33:32
If A was the zero matrix, of
course, we would just get the
536
00:33:32 --> 00:33:37
zero matrix which would be only
semi-definite, or other ways
537
00:33:37 --> 00:33:42
to get a semi-definite.
538
00:33:42 --> 00:33:46
So I'm saying that if K, if I
have a matrix, any matrix, and
539
00:33:46 --> 00:33:51
I form A transpose A, I get a
positive definite matrix or
540
00:33:51 --> 00:33:56
maybe just semi-definite,
but not indefinite.
541
00:33:56 --> 00:34:01
Can we see why?
542
00:34:01 --> 00:34:11
Why is this positive
definite or semi-?
543
00:34:11 --> 00:34:13
So that's my question.
544
00:34:13 --> 00:34:17
And the answer is really
worth, it's just neat
545
00:34:17 --> 00:34:19
and worth seeing.
546
00:34:19 --> 00:34:23
So do I want to look at the
pivots of A transpose A?
547
00:34:23 --> 00:34:25
No.
548
00:34:25 --> 00:34:29
They're something, but whatever
they are, I can't really
549
00:34:29 --> 00:34:30
follow those well.
550
00:34:30 --> 00:34:34
Or the eigenvalues very
well, or the determinants.
551
00:34:34 --> 00:34:36
None of those come out nicely.
552
00:34:36 --> 00:34:41
But the real guy
works perfectly.
553
00:34:41 --> 00:34:46
So look at x transpose Kx.
554
00:34:46 --> 00:34:48
555
00:34:48 --> 00:34:57
So I'm just doing, following
my instinct here.
556
00:34:57 --> 00:35:03
So if K is A transpose A, my
claim is, what am I saying
557
00:35:03 --> 00:35:07
then about this energy?
558
00:35:07 --> 00:35:13
What is it that I want to
discover and understand?
559
00:35:13 --> 00:35:15
Why it's positive.
560
00:35:15 --> 00:35:20
Why does taking any matrix,
multiplying by its transpose
561
00:35:20 --> 00:35:27
produce something
that's positive?
562
00:35:27 --> 00:35:31
Can you see any reason why that
quantity, which looks kind of
563
00:35:31 --> 00:35:38
messy, I just want to look at
it the right way to see why
564
00:35:38 --> 00:35:41
that should be positive, that
should come out positive.
565
00:35:41 --> 00:35:45
So I'm not going to get into
numbers, I'm not going to get
566
00:35:45 --> 00:35:47
into diagonals and
off-diagonals.
567
00:35:47 --> 00:35:53
I'm just going to do one thing
to understand that particular
568
00:35:53 --> 00:35:56
combination, x transpose
A transpose Ax.
569
00:35:57 --> 00:35:59
What shall I do?
570
00:35:59 --> 00:36:06
Anybody see what I might do?
571
00:36:06 --> 00:36:10
Yeah, you're seeing here
if you look at it again,
572
00:36:10 --> 00:36:12
what are you seeing here?
573
00:36:12 --> 00:36:14
Tell me again.
574
00:36:14 --> 00:36:20
If I take Ax together, then
what's the other half?
575
00:36:20 --> 00:36:23
It's the transpose of Ax.
576
00:36:23 --> 00:36:26
So I just want to write that
as, I just want to think
577
00:36:26 --> 00:36:27
of it that way, as Ax.
578
00:36:29 --> 00:36:31
And here's the transpose of Ax.
579
00:36:32 --> 00:36:33
Right?
580
00:36:33 --> 00:36:36
Because transposes of Ax, so
transpose guys in the opposite
581
00:36:36 --> 00:36:39
order, and the multiplication--
582
00:36:39 --> 00:36:41
This is the great.
583
00:36:41 --> 00:36:44
I call these proof by
parenthesis because I'm just
584
00:36:44 --> 00:36:51
putting parentheses in the
right place, but the key law
585
00:36:51 --> 00:36:57
of matrix multiplication is
that, that I can put (AB)C
586
00:36:57 --> 00:36:58
is the same as A(BC).
587
00:36:58 --> 00:37:01
588
00:37:01 --> 00:37:04
That rule, which is just
multiply it out and you see
589
00:37:04 --> 00:37:07
that parentheses are not needed
because if you keep them in the
590
00:37:07 --> 00:37:10
right order you can do this
first, or you can
591
00:37:10 --> 00:37:12
do this first.
592
00:37:12 --> 00:37:13
Same answer.
593
00:37:13 --> 00:37:15
What do I learn from that?
594
00:37:15 --> 00:37:17
What was the point?
595
00:37:17 --> 00:37:19
This is some vector, I don't
know especially what it
596
00:37:19 --> 00:37:21
is times its transpose.
597
00:37:21 --> 00:37:24
So that's the length squared.
598
00:37:24 --> 00:37:27
What's the key fact about that?
599
00:37:27 --> 00:37:30
That it is never negative.
600
00:37:30 --> 00:37:41
It's always greater than
zero or possibly equal.
601
00:37:41 --> 00:37:44
When does that
quantity equal zero?
602
00:37:44 --> 00:37:45
When Ax is zero.
603
00:37:45 --> 00:37:47
When Ax is zero.
604
00:37:47 --> 00:37:49
Because this is a vector.
605
00:37:49 --> 00:37:50
That's the same
vector transposed.
606
00:37:50 --> 00:37:52
And everybody's
got that picture.
607
00:37:52 --> 00:37:58
When I take any y transpose y,
I get y_1 squared plus y_2
608
00:37:58 --> 00:38:00
squared through y_n squared.
609
00:38:00 --> 00:38:05
And I get a positive answer
except if the vector is zero.
610
00:38:05 --> 00:38:11
So it's zero when Ax is zero.
611
00:38:11 --> 00:38:13
So that's going to be the key.
612
00:38:13 --> 00:38:19
If I pick any matrix A, and I
can just take an example, but
613
00:38:19 --> 00:38:22
chapter, the applications
are just going to be
614
00:38:22 --> 00:38:23
full of examples.
615
00:38:23 --> 00:38:29
Where the problem begins with a
matrix A and then A transpose
616
00:38:29 --> 00:38:34
shows up and it's the
combination A transpose
617
00:38:34 --> 00:38:36
A that we work with.
618
00:38:36 --> 00:38:40
And we're just learning that
it's positive definite.
619
00:38:40 --> 00:38:48
Unless, shall I just hang on
since I've got here, I have to
620
00:38:48 --> 00:38:53
say when is it, have to get
these two possibilities.
621
00:38:53 --> 00:38:56
Positive definite or
only semi-definite.
622
00:38:56 --> 00:39:05
So what's the key to that
borderline question?
623
00:39:05 --> 00:39:11
This thing will be only
semi-definite if there's
624
00:39:11 --> 00:39:12
a solution to Ax=0.
625
00:39:12 --> 00:39:16
626
00:39:16 --> 00:39:23
If there is an x, well, there's
always the zero vector.
627
00:39:23 --> 00:39:26
Zero vector I can't
expect to be positive.
628
00:39:26 --> 00:39:35
So I'm looking for if there's
an x so that Ax is zero but x
629
00:39:35 --> 00:39:48
is not zero, then I'll
only be semi-definite.
630
00:39:48 --> 00:39:50
That's the test.
631
00:39:50 --> 00:39:52
If there is a solution to Ax=0.
632
00:39:52 --> 00:39:55
633
00:39:55 --> 00:39:59
When we see applications
that'll mean there's a
634
00:39:59 --> 00:40:03
displacement with
no stretching.
635
00:40:03 --> 00:40:10
We might have a line of springs
and when could the line
636
00:40:10 --> 00:40:16
of springs displace
with no stretching?
637
00:40:16 --> 00:40:18
When it's free-free, right?
638
00:40:18 --> 00:40:24
If I have a line of springs and
no supports at the ends, then
639
00:40:24 --> 00:40:27
that would be the case where
it could shift over by
640
00:40:27 --> 00:40:29
the vector.
641
00:40:29 --> 00:40:33
So that would be the case where
the matrix is only singular.
642
00:40:33 --> 00:40:34
We know that.
643
00:40:34 --> 00:40:37
The matrix is now
positive semi-definite.
644
00:40:37 --> 00:40:38
We just learned that.
645
00:40:38 --> 00:40:46
So the free-free matrix, like
B, both ends free, or C.
646
00:40:46 --> 00:40:52
So our answer is going
to be that K and T are
647
00:40:52 --> 00:40:56
positive definite.
648
00:40:56 --> 00:40:59
And our other two guys, the
singular ones, of course,
649
00:40:59 --> 00:41:00
just don't make it.
650
00:41:00 --> 00:41:04
B at both ends, the free-free
line of springs, it can
651
00:41:04 --> 00:41:07
shift without stretching.
652
00:41:07 --> 00:41:11
Since Ax will measure the
stretching when it just shifts
653
00:41:11 --> 00:41:14
rigid motion, the Ax is
zero and we see only
654
00:41:14 --> 00:41:16
positive definite.
655
00:41:16 --> 00:41:19
And also C, the circular one.
656
00:41:19 --> 00:41:22
There it can displace with no
stretching because it can
657
00:41:22 --> 00:41:24
just turn in the circle.
658
00:41:24 --> 00:41:45
So these guys will be only
positive semi-definite.
659
00:41:45 --> 00:41:49
Maybe I better say
this another way.
660
00:41:49 --> 00:41:51
When is this positive definite?
661
00:41:51 --> 00:41:55
Can I use just a different
sentence to describe
662
00:41:55 --> 00:41:57
this possibility?
663
00:41:57 --> 00:42:05
This is positive definite
provided, so what I'm going to
664
00:42:05 --> 00:42:08
write now is to remove this
possibility and get
665
00:42:08 --> 00:42:10
positive definite.
666
00:42:10 --> 00:42:16
This is positive definite
provided, now, I could
667
00:42:16 --> 00:42:17
say it this way.
668
00:42:17 --> 00:42:25
The A has independent columns.
669
00:42:25 --> 00:42:28
So I just needed to give
you another way of looking
670
00:42:28 --> 00:42:33
at this Ax=0 question.
671
00:42:33 --> 00:42:37
If A has independent columns,
what does that mean?
672
00:42:37 --> 00:42:40
That means that the only
solution to Ax=0 is
673
00:42:40 --> 00:42:42
the zero solution.
674
00:42:42 --> 00:42:47
In other words, it means that
this thing works perfectly
675
00:42:47 --> 00:42:50
and gives me positive.
676
00:42:50 --> 00:42:53
When A has independent columns.
677
00:42:53 --> 00:42:56
Let's just remember
our K, T, B, C.
678
00:42:56 --> 00:43:08
So here's a matrix, so let me
take the T matrix, that's
679
00:43:08 --> 00:43:11
this one, this guy.
680
00:43:11 --> 00:43:15
And then the third
column is .
681
00:43:15 --> 00:43:19
Those three columns
are independent.
682
00:43:19 --> 00:43:21
They point off.
683
00:43:21 --> 00:43:23
They don't lie in a plane.
684
00:43:23 --> 00:43:27
They point off in three
different directions.
685
00:43:27 --> 00:43:34
And then there are no solutions
to, no x's that's go Kx=0.
686
00:43:34 --> 00:43:39
687
00:43:39 --> 00:43:41
So that would be a case
of independent columns.
688
00:43:41 --> 00:43:45
Let me make a case of
dependent columns.
689
00:43:45 --> 00:43:47
So and I'm going
to make it B now.
690
00:43:47 --> 00:43:51
Now the columns of that
matrix are dependent.
691
00:43:51 --> 00:43:54
There's a combination of
them that give zero.
692
00:43:54 --> 00:43:56
They all lie in the same plane.
693
00:43:56 --> 00:44:00
There's a solution to that
matrix times x equal zero.
694
00:44:00 --> 00:44:03
What combination of those
columns shows me that
695
00:44:03 --> 00:44:05
they are dependent?
696
00:44:05 --> 00:44:09
That some combination of those
three columns, some amount of
697
00:44:09 --> 00:44:12
this plus some amount of this
plus some amount of that column
698
00:44:12 --> 00:44:15
gives me the zero vector.
699
00:44:15 --> 00:44:17
You see the combination.
700
00:44:17 --> 00:44:21
What should I take?
again.
701
00:44:21 --> 00:44:22
No surprise.
702
00:44:22 --> 00:44:27
That's the vector
that we know is in the
703
00:44:27 --> 00:44:36
everything shifting the same
amount, nothing stretching.
704
00:44:36 --> 00:44:40
Talking fast here about
positive definite matrices.
705
00:44:40 --> 00:44:42
This is the key.
706
00:44:42 --> 00:44:44
Let's just ask a few questions
about positive definite
707
00:44:44 --> 00:44:49
matrices as a way to practice.
708
00:44:49 --> 00:44:50
Suppose I had one.
709
00:44:50 --> 00:44:52
Positive definite.
710
00:44:52 --> 00:44:57
What about its inverse?
711
00:44:57 --> 00:45:02
Is that positive
definite or not?
712
00:45:02 --> 00:45:06
So I've got a positive definite
one, it's not singular, it's
713
00:45:06 --> 00:45:09
got positive eigenvalues,
everything else.
714
00:45:09 --> 00:45:14
It's inverse will be
symmetric, so I'm allowed
715
00:45:14 --> 00:45:16
to think about it.
716
00:45:16 --> 00:45:20
Will it be positive definite?
717
00:45:20 --> 00:45:23
What do you think?
718
00:45:23 --> 00:45:27
Well, you've got a whole
bunch of tests to sort
719
00:45:27 --> 00:45:30
of mentally run through.
720
00:45:30 --> 00:45:35
Pivots of the inverse, you
don't want to touch that stuff.
721
00:45:35 --> 00:45:36
Determinants, no.
722
00:45:36 --> 00:45:39
What about eigenvalues?
723
00:45:39 --> 00:45:42
What would be the eigenvalues
if I have this positive
724
00:45:42 --> 00:45:44
definite symmetric matrix,
its eigenvalues are
725
00:45:44 --> 00:45:46
one, four, five.
726
00:45:46 --> 00:45:49
What can you tell me
about the eigenvalues
727
00:45:49 --> 00:45:53
of the inverse matrix?
728
00:45:53 --> 00:45:54
They're the inverses.
729
00:45:54 --> 00:45:56
So those three eigenvalues are?
730
00:45:56 --> 00:46:00
1, 1/4, 1/5, what's
the conclusion here?
731
00:46:00 --> 00:46:02
It is positive definite.
732
00:46:02 --> 00:46:04
Those are all positive,
it is positive definite.
733
00:46:04 --> 00:46:08
So if I invert a positive
definite matrix, I'm
734
00:46:08 --> 00:46:11
still positive definite.
735
00:46:11 --> 00:46:13
All the tests would
have to pass.
736
00:46:13 --> 00:46:17
It's just I'm looking each
time for the easiest test.
737
00:46:17 --> 00:46:22
Let me look now, for the
easiest test on K_1+K_2.
738
00:46:22 --> 00:46:25
739
00:46:25 --> 00:46:27
Suppose that's positive
definite and that's
740
00:46:27 --> 00:46:29
positive definite.
741
00:46:29 --> 00:46:33
What if I add them?
742
00:46:33 --> 00:46:35
What do you think?
743
00:46:35 --> 00:46:38
Well, we hope so.
744
00:46:38 --> 00:46:42
But we have to say which of my
one, two, three, four, five
745
00:46:42 --> 00:46:45
would be a good way to see it.
746
00:46:45 --> 00:46:48
Would be a good way to see it.
747
00:46:48 --> 00:46:50
Good question.
748
00:46:50 --> 00:46:53
Four?
749
00:46:53 --> 00:46:55
We certainly don't want to
touch pivots and now we
750
00:46:55 --> 00:46:58
don't want to touch
eigenvalues either.
751
00:46:58 --> 00:47:03
Of course, if number four
works, others will also work.
752
00:47:03 --> 00:47:05
The eigenvalues will
come out positive.
753
00:47:05 --> 00:47:08
But not too easy to
say what they are.
754
00:47:08 --> 00:47:14
Let's try test number four.
755
00:47:14 --> 00:47:15
So K_1.
756
00:47:15 --> 00:47:18
757
00:47:18 --> 00:47:20
What's the test?
758
00:47:20 --> 00:47:23
So test number four tells us
that this part, x transpose
759
00:47:23 --> 00:47:28
K_1*x, that that part
is positive, right?
760
00:47:28 --> 00:47:30
That that part is positive.
761
00:47:30 --> 00:47:33
If we know that's
positive definite.
762
00:47:33 --> 00:47:37
Now, about K_2 we also know
that for every x, you see it's
763
00:47:37 --> 00:47:42
for every x, that helps, don't
let me put x_2 there, for every
764
00:47:42 --> 00:47:47
x this will be positive.
765
00:47:47 --> 00:47:52
And now what's the
step I want to take?
766
00:47:52 --> 00:47:57
To get some information
on the matrix K_1+K_2.
767
00:47:57 --> 00:47:59
768
00:47:59 --> 00:48:01
I should add.
769
00:48:01 --> 00:48:07
If I add these guys, you see
that it just, then I can
770
00:48:07 --> 00:48:14
write that as, I can
write that this way.
771
00:48:14 --> 00:48:17
And what have I learned?
772
00:48:17 --> 00:48:19
I've learned that that's
positive, even greater than,
773
00:48:19 --> 00:48:21
except for the zero vector.
774
00:48:21 --> 00:48:23
Because this was greater
than, this is greater than.
775
00:48:23 --> 00:48:27
If I add two positive numbers,
the energies are positive
776
00:48:27 --> 00:48:29
and the energies just add.
777
00:48:29 --> 00:48:34
The energies just add.
778
00:48:34 --> 00:48:40
So that definition four was the
good way, just nice, easy way
779
00:48:40 --> 00:48:44
to see that if I have a couple
of positive definite matrices,
780
00:48:44 --> 00:48:47
a couple of positive energies,
I'm really coupling
781
00:48:47 --> 00:48:49
the two systems.
782
00:48:49 --> 00:48:53
This is associated somehow.
783
00:48:53 --> 00:48:55
I've got two systems, I'm
putting them together
784
00:48:55 --> 00:49:00
and the energy is just
even more positive.
785
00:49:00 --> 00:49:05
It's more positive either of
these guys because I'm adding.
786
00:49:05 --> 00:49:11
As I'm speaking here, will you
allow me to try test number
787
00:49:11 --> 00:49:14
five, this A transpose
A business?
788
00:49:14 --> 00:49:20
Suppose K_1 was A transpose A.
789
00:49:20 --> 00:49:21
If it's positive
definite, it will.
790
00:49:21 --> 00:49:31
Be And suppose K_2
is B transpose B.
791
00:49:31 --> 00:49:33
If it's positive
definite, it will be.
792
00:49:33 --> 00:49:42
Now I would like to write the
sum somehow as, in this
793
00:49:42 --> 00:49:43
something transpose something.
794
00:49:43 --> 00:49:47
And I just do it now because
I think it's like, you
795
00:49:47 --> 00:49:54
won't perhaps have thought
of this way to do it.
796
00:49:54 --> 00:49:56
Watch.
797
00:49:56 --> 00:50:01
Suppose I create
the matrix [A; B].
798
00:50:01 --> 00:50:03
That'll be my new matrix.
799
00:50:03 --> 00:50:08
Say, call it C.
800
00:50:08 --> 00:50:11
Am I allowed to do that?
801
00:50:11 --> 00:50:13
I mean, that creates a matrix?
802
00:50:13 --> 00:50:18
These A and B, they had the
same number of columns, n.
803
00:50:18 --> 00:50:20
So I can put one over the
other and I still have
804
00:50:20 --> 00:50:22
something with n columns.
805
00:50:22 --> 00:50:24
So that's my new matrix C.
806
00:50:24 --> 00:50:26
And now I want C transpose.
807
00:50:26 --> 00:50:31
By the way, I'd call
that a block matrix.
808
00:50:31 --> 00:50:35
You know, instead of numbers,
it's got two blocks in there.
809
00:50:35 --> 00:50:37
Block matrices are
really handy.
810
00:50:37 --> 00:50:43
Now what's the transpose
of that block matrix?
811
00:50:43 --> 00:50:47
You just have faith, just
have faith with blocks.
812
00:50:47 --> 00:50:48
It's just like numbers.
813
00:50:48 --> 00:50:55
If I had a matrix [1; 5] then
I'd get a row one, five.
814
00:50:55 --> 00:50:57
But what do you think?
815
00:50:57 --> 00:51:01
This is worth thinking
about even after class.
816
00:51:01 --> 00:51:05
What would be, if this C matrix
is this block A above B, what
817
00:51:05 --> 00:51:07
do you think for C transpose?
818
00:51:07 --> 00:51:11
A transpose, B transpose
side by side.
819
00:51:11 --> 00:51:15
Just put in numbers
and you'd see it.
820
00:51:15 --> 00:51:19
And now I'm going to take
C transpose times C.
821
00:51:19 --> 00:51:25
I'm calling it C now instead of
A because I've used the A in
822
00:51:25 --> 00:51:27
the first guy and I've used B
in the second one and
823
00:51:27 --> 00:51:31
now I'm ready for C.
824
00:51:31 --> 00:51:35
How do you multiply
block matrices?
825
00:51:35 --> 00:51:37
Again, you just have faith.
826
00:51:37 --> 00:51:39
What do you think?
827
00:51:39 --> 00:51:41
Tell me the answer.
828
00:51:41 --> 00:51:44
A transpose, I multiply
that by that just as
829
00:51:44 --> 00:51:47
if they were numbers.
830
00:51:47 --> 00:51:52
And I add that times that just
as if they were numbers.
831
00:51:52 --> 00:51:55
And what do I have?
832
00:51:55 --> 00:51:55
I've got K_1+K_2.
833
00:51:55 --> 00:51:58
834
00:51:58 --> 00:52:05
So I've written K_1, this is
K_1+K_2 and this is in my form
835
00:52:05 --> 00:52:08
C transpose C that I was
looking for, that number
836
00:52:08 --> 00:52:10
five was looking for.
837
00:52:10 --> 00:52:12
So it's done it.
838
00:52:12 --> 00:52:13
It's done it.
839
00:52:13 --> 00:52:19
The fact of getting A, K_1 in
this form, K_2 in this form.
840
00:52:19 --> 00:52:21
And I just made a block
matrix and I got K_1+K_2.
841
00:52:21 --> 00:52:25
842
00:52:25 --> 00:52:29
That's not a big deal in
itself, but block matrices
843
00:52:29 --> 00:52:32
are really handy.
844
00:52:32 --> 00:52:36
It's good to take that
step with matrices.
845
00:52:36 --> 00:52:40
Think of, possibly, the entries
as coming in blocks and
846
00:52:40 --> 00:52:42
not just one at a time.
847
00:52:42 --> 00:52:44
Well, thank you, okay.
848
00:52:44 --> 00:52:51
I swear Friday we'll start
applications in all kinds of
849
00:52:51 --> 00:52:55
engineering problems and
you'll have new applications.
850
00:52:55 --> 00:52:55