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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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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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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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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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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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And knowing that that
isn't maybe a good

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hour for everybody.
 

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So you could see him also
outside of that hour.

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But that's the hour he will
be ready for all kinds of

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questions about MATLAB
or about the homeworks.

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Actually you'll be probably
thinking more also about the

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homework questions
on this topic.

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Ready for positive definite?
 

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You said yes, right?
 

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And you have a hint
about these things.

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So we have a symmetric matrix
and the beauty is that it

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brings together all
of linear algebra.

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Including elimination,
that's when we see pivots.

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Including determinants
which are closely

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related to the pivots.
 

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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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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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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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And of course if I give you a
three by three matrix, that's

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probably not the easiest
test since you'd have to

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find the eigenvalues.
 

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Much easier to find the
determinants or the pivots.

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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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a*c - b squared to a.
 

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So the connection between
pivots and determinants

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is just really close.
 

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Pivots are ratios
of determinants if

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you work it out.
 

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The second pivot, maybe I would
call that d_2, is the ratio

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of a*c - b squared over a.
 

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In other words it's
(c - b squared)/a.

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00:05:41 --> 00:05:48
 
 

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Determinants are positive
and vice versa.

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Then it's fantastic that
the eigenvalues come

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into the picture.
 

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So those are three ways, three
important properties of a

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positive definite matrix.
 

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But I'd like to make the
definition something different.

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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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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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And so what I'm going to bring
in here, so it's symmetric.

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Now for a symmetric matrix I
want to introduce the energy.

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So this is the reason why it
has so many applications and

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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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transpose times A, not A, I'm
sticking with K for

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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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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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I've ever written something
that has x's times x's.

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So it's going to be quadratic.
 

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They're going to
be x's times x's.

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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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And there it's transpose,
so it's a row.

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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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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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So here, that would
be ax_1+bx_2.

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00:08:30 --> 00:08:35
 
 

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And this would be bx_1+cx_2.
 

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00:08:36 --> 00:08:39
 
 

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So that's the first,
that's this times this.

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What am I going to get?
 

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What shape, what size is
this result going to be?

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This K is n by n. x is a column
vector. n by one. x transpose,

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what's the shape
of x transpose?

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One by n?
 

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So what's the total result?
 

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One by one.
 

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Just a number.
 

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Just a function.
 

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It's a number.
 

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But it depends on the x's
and the matrix inside.

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Can we do it now?
 

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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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Yes, from there times there.
 

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And what's it multiplied by?
 

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The a.
 

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The first term is this times
the ax_1 is a(x_1 squared).

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So that's our first quadratic.
 

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Now there'd be an x_1, x_2.
 

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Let me leave that for a minute
and find the x_2 squared

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because it's easy.
 

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So where am I going
to get x_2 squared?

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I'm going to get that from
x_2, second guy here

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times second guy here.
 

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There's a c(x_2 squared).
 

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So you're seeing already
where the diagonal shows up.

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The diagonal a, c, whatever
is multiplying the

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perfect squares.
 

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And it'll be the off-diagonal
that multiplies the x_1, x_2.

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We might call those
the crossterms.

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And what do we get
for that then?

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We have x_1 times this guy.
 

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So that's a crossterm.
bx_1*x_2, right?

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And here's another one coming
from x_2 times this guy.

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And what's that one?
 

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It's also bx_1*x_2.
 

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So x_1, multiply that, x_2
multiply that, and so what

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do we have for this
crossterm here?

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Two of them.
 

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2bx_1*x_2.
 

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In other words, that b
and that b came together

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in the 2bx_1*x_2.
 

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So here's my energy.
 

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Can I just loosely
call it energy?

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And then as we get to
applications we'll see why.

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So I'm interested in
that because it has

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important meaning.
 

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Well, so now I'm ready
to define positive

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definite matrices.
 

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So I'll call that number four.
 

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But I'm going to
give it a big star.

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Even more because it's
the sort of key.

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So the test will be, you can
probably guess it, I look at

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this expression, that
x transpose Ax.

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And if it's a positive definite
matrix and this represents

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energy, the key will be that
this should be positive.

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This one should be
positive for all x's.

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Well, with one
exception, of course.

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All x's except, which vector
is it? x=0 would just

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give me-- See, I put K.
 

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My default for a matrix,
but should be, it's K.

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Except x=0, except
the zero vector.

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Of course.
 

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If x_1 and x_2 are both zero.
 

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Now that looks a little maybe
less straightforward, I would

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say, because it's a statement
about this is true

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for all x_1 and x_2.
 

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And we better do some
examples and draw a picture.

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Let me draw a
picture right away.

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So here's x_1 direction.
 

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Here's x_2 direction.
 

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And here is the x transpose
Ax, my function.

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00:13:05 --> 00:13:09
So this depends on
two variables.

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00:13:09 --> 00:13:13
So it's going to be a sort
of a surface if I draw it.

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00:13:13 --> 00:13:16
Now, what point do
we absolutely know?

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And I put A again.
 

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00:13:21 --> 00:13:29
I am so sorry.
 

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00:13:29 --> 00:13:30
Well, we know one point.
 

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It's there whatever
that matrix might be.

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00:13:34 --> 00:13:35
It's there.
 

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00:13:35 --> 00:13:37
Zero, right?
 

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00:13:37 --> 00:13:40
You just told me that if
both x's are zero then we

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00:13:40 --> 00:13:42
automatically get zero.
 

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00:13:42 --> 00:13:47
Now what do you think the shape
of this curve, the shape of

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00:13:47 --> 00:13:51
this graph is going
to look like?

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The point is, if we're
positive definite now.

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00:13:54 --> 00:13:58
So I'm drawing the picture
for positive definite.

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So my definition is that
the energy goes up.

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00:14:04 --> 00:14:06
It's positive, right?
 

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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.

234
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
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That's why minimization, which
was like, the core problem

240
00:14:46 --> 00:14:49
in calculus, is here now.
 

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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.
 

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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