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PROFESSOR STRANG: This is
my second lecture on the

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example, the model of
graphs and networks.

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Really perfect, beautiful model
for the whole framework.

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And important in itself.
 

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So, I guess a major thing
that I still have to do

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is discuss A transpose.
 

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See how A transpose just
naturally appears in

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the balance slope.
 

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Kirchhoff's current law,
KCL, is just like a model

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for balance equations.
 

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By balance I mean, n
equals out, essentially.

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Flow in equals flow out
because we're talking

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about steady state.
 

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So in each node, at node
three, for example,

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I have three edges.
 

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So the law, Kirchhoff's current
law at that point is going to

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tell me that the flow w_2 plus
w_3 minus w_5 would be zero if

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there's no current source, or
if I'm feeding current into

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there, some current f_3,
then it would match f_3.

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So maybe having just said those
words, let me just say w_2,

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I'll say it again, w_2 plus w_3
minus w_5, and I'm hoping that

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I'll find that in the
third column here.

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And I do.
 

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Because I'm thinking, and we'll
write it down, I'm thinking to

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take a transpose, so it'll be
the third row, and here I see

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that w_2, w_3 and minus w_5
that will show up in the

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third equation there, the n
equals out at node three.

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Well I'll write that down,
because it's -- So, two

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or three jobs for today.
 

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And then Monday I plan to spend
a part of the lecture and all

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of the review session in
review, it's a great chance to

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go back to the things we've
done and collect them, assemble

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them, organize them
and see them again.

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So that's my goal for
both sessions on Monday.

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And questions, then,
about every aspect.

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So, before I get to A
transpose, this part I had

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written down last time, but
there's a little more to say.

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And what I was saying and
want to continue with

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is the source terms.
 

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I put in this b but I didn't
draw anything on the network.

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So let me draw what
these b's represent.

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So this second, lower, row
is about edge equations,

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edge variables.
 

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So those batteries b
are on the edges.

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And there will be,
b has length five.

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There are five edges.
 

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This is a vector of length
five, this matrix, A, you

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remember was five by four.
 

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It produced from four inputs,
from four potentials at the

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four nodes, A produces Au,
five potential differences.

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So those are the
differences in potentials.

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But then also there can be
source terms from batteries in

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the edges, and the minus sign
is there because I'm talking

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about voltage drops.
 

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So when I say differences
I'm really speaking

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about voltage drops.
 

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Because that's the
way current flows.

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OK, this is the
moment that I hate.

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Putting the batteries in I
think you draw a battery

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with a long and a short?
 

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Is that right?
 

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And then you put a
plus and a minus?

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Well, can I just say, life
is too short to get those,

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to get this sign right.
 

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You may say when my car battery
stalls how do I, because it's

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important at that
moment, right?

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When you start it up you're
supposed to put the right

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battery, the right lead on the
positive and the right on the

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negative, or your
blow yourself up.

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OK.
 

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So what's my solution?
 

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Because I literally refuse
to deal with these signs.

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So my solution is call AAA.
 

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OK, they're paid to
blow themselves up.

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They can do it.
 

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But so this is serious now,
I don't want to be asked

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about these signs.
 

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And I forgive you for
messing up the signs.

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So possibly that's plus,
possibly minus, I don't know.

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But there's a battery in there
of length b_1, of voltage, of a

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nine-volt battery,
b_1 would be nine.

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And depending how it was placed
in there, the b here would be a

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plus nine or a minus nine
in the first component.

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And then if I had another
battery here, a b that's on

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edge four, there would be a
battery b_4, and that would

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show up there in Ohm's law.
 

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Because Ohm's law will look
at the difference in these

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potentials, but then it'll also
account for the battery, right?

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The voltage that comes from the
battery, and somehow combining

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the Au, which is the difference
in these guys, with the b_4,

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I'll know what is the voltage
drop across the resistor.

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And that's what Ohm's
law applies to.

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Ohm's law says the voltage drop
across the resistor times the

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conductance, so this is Ohm's
law, that on that edge the

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voltage drop across the
resistor times the conductance

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gives the current.
 

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So that's the physical law.
 

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Is that OK?
 

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For batteries.
 

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Now a comment on
current sources.

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00:07:12 --> 00:07:14
So what's with current sources?
 

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How would I draw those?
 

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Well, very often maybe I might
draw a current going into node

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one, so that would be in f_1
that goes into node one and

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maybe comes out at ground.
 

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So that would be a typical f.
 

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That if I imposed a current
source, if I sent a current

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source through there it would
go down and come out at ground,

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and our question is what are
the currents in the five edges?

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What are the potentials
at the four nodes?

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Well, I'm making
this one ground.

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So I'm grounding this
one to be u_4=0.

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And you remember of course why
I had to do that, because this

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matrix a transpose a, as it
stands is, what's the matter

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with a transpose
a as it stands?

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It's singular, right?
 

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And as I, looking ahead, just a
small comment, that this will

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have exactly the same thing in
so many other applications in

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big, finite element codes, you
produce a stiffness matrix

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that's initially singular.
 

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And then you impose the
boundary conditions.

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That's the efficient way to do
it, is create the matrix first,

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that's the big job, then impose
the boundary conditions, that's

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the small but occasionally
tricky part.

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Now I indicated what
happened here.

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When u_4 was zero, that means
that a times u, the u_4 there,

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the zero is multiplying this,
is not unknown any more.

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It's known.
 

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And so that column is not
really any more leaded in A.

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Because u_4 is gone from my
list of unknown potentials.

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Now, at the same time, when I
went over to A transpose A, I'm

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making this comment because
there were several good

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questions about it.
 

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I claim that also, that row
coming, which of course

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comes from the fourth
row of A transpose.

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The fourth column of A is
gone, so the fourth row A

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transpose should be gone.
 

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And we might just think why's
that, what's going on?

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Of course, it produces
exactly what we want.

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It leaves us with a three
by three matrix that's

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not singular any more.
 

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I've removed that <1, 1,
1, 1> from the null space

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by fixing a potential.
 

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Grounding a node, and
the problem becomes

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just what we want.
 

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00:10:05 --> 00:10:10
And I'll write down the
equations when I get the

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Kirchhoff current law
to complete the loop.

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Now, what's going on?
 

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Let me remove this current
source, just so we focus on

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what I'm speaking about now.
 

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On this type of boundary
condition, this is like

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fixing a support, right?
 

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It's like fixing a support
in our spring mass problem.

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Can I squeeze in a little
spring mass problem, so I have

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a spring, and then I'm fixing
this displacement,

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u_4 to be zero.
 

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Now, I want to try to think
through what happens in a

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balance equation, A transpose
w, let me make it A transpose

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w equal f, because that's
the case with right

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hand side allowed.
 

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And now what happens when I fix
this, I'm asking you to think

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back about force balance, which
we certainly saw was an A

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transpose w equal f business.
 

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And then parallel will be the
current balance, at that node.

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OK, so what was the
deal on force balance?

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A transpose for this
fixed in here.

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What was the thing with a
transpose did that fixed in?

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We did not write the equation
A transpose w equal

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f at this point.
 

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We did not write the force
balance equation at that point.

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00:12:14 --> 00:12:19
When I fixed u_4, in this case
it was the displacement so I'm

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fixing it at zero displacement,
I didn't have a force balance

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in the A transpose
part at this node.

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Now, you could say why?
 

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Because of course forces
have to balance.

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But what's going on?
 

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What's really happening here is
I don't have to write out, I

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the displacement here is known.
 

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It's not unknown.
 

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And let me say it
in a sentence.

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Force balance does hold because
the support supplies whatever

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force it takes to balance
the internal forces.

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00:13:01 --> 00:13:09
So, in other words, let me say
that again, the force balance

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will hold and it will tell me,
after I've solved the rest of

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the problem, it'll tell me what
the support has to supply, how

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much force the support
is actually supplying.

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It's a reaction force,
it would be called.

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So a reaction force is whatever
the support has to do to

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fix that displacement.
 

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And so I solved the fixed
problems for all the other

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displacements and all
the spring forces.

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And then I could come back at
the end and say OK, what was

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the force in that spring and
therefore how much is that

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support, what's the force being
supplied by that support.

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00:13:52 --> 00:13:53
See what I'm saying?
 

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That the force balance at
this node comes afterwards.

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00:14:05 --> 00:14:12
That equation is like knocking
that one out of this problem.

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00:14:12 --> 00:14:17
It would be the same here I
fix the potential at zero.

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00:14:17 --> 00:14:19
I fix ground at zero.
 

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Current flows.
 

209
00:14:21 --> 00:14:26
Maybe some, maybe I might
fix that potential at one.

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00:14:26 --> 00:14:28
I fix that potential at zero.
 

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00:14:28 --> 00:14:29
Current flows.
 

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00:14:29 --> 00:14:36
OK. ta-da, I find out, I
compute what they are, and this

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00:14:36 --> 00:14:40
row and column will be gone.
 

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00:14:40 --> 00:14:41
Find out what they are.
 

215
00:14:41 --> 00:14:46
At this node, what's happened?
 

216
00:14:46 --> 00:14:50
What's happening to the
balance of currents?

217
00:14:50 --> 00:14:52
Does this current
have to balance that

218
00:14:52 --> 00:14:54
current at the ground?
 

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00:14:54 --> 00:14:55
No.
 

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The ground, whatever current
comes here plus whatever

221
00:15:01 --> 00:15:04
current comes here,
goes into ground.

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00:15:04 --> 00:15:07
Do you see the point?
 

223
00:15:07 --> 00:15:12
Kirchhoff's current law, the
balance of currents that in

224
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equals out, is true but it's
not an equation I have to solve

225
00:15:18 --> 00:15:21
in finding the currents,
it's something I can

226
00:15:21 --> 00:15:22
discover at the end.
 

227
00:15:22 --> 00:15:25
I can say OK, how much
current flowed from ground?

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00:15:25 --> 00:15:27
And similarly up here.
 

229
00:15:27 --> 00:15:32
If I fix u_1 to be zero, then
you, no, I don't want to fix it

230
00:15:32 --> 00:15:36
to be zero, two that would be a
way to make very little happen.

231
00:15:36 --> 00:15:43
Let me fix u_1 to be one,
so this is a standard

232
00:15:43 --> 00:15:44
important problem.
 

233
00:15:44 --> 00:15:49
It's like what's the resistance
in the net, and what's the

234
00:15:49 --> 00:15:50
net system resistance?
 

235
00:15:50 --> 00:15:55
If I fix this at one fix this
at zero, some current will

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flow, it'll come out here.
 

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And that'll be the
current going into here.

238
00:16:01 --> 00:16:06
Somehow there'll be a balance
there, and a balance there.

239
00:16:06 --> 00:16:10
But it's found later.
 

240
00:16:10 --> 00:16:14
Just the way the force in
the support is found later.

241
00:16:14 --> 00:16:16
OK, that takes a
little thinking.

242
00:16:16 --> 00:16:23
I just wanted to, because I had
blithely knocked this row out.

243
00:16:23 --> 00:16:27
And you could do it on the
basis, well if you knock out

244
00:16:27 --> 00:16:30
this column of a then you're
knocking out the row of A

245
00:16:30 --> 00:16:33
transpose, and therefore A
transpose A will be

246
00:16:33 --> 00:16:35
three by three.
 

247
00:16:35 --> 00:16:40
And it'll go down to two by
two if I fix that potential.

248
00:16:40 --> 00:16:43
And if I don't fix it at zero,
if I fix it at one then

249
00:16:43 --> 00:16:47
something will move to
the right hand side.

250
00:16:47 --> 00:16:57
OK. that's that point I
hope at least discussed.

251
00:16:57 --> 00:17:05
So now we have the whole
pattern here. except that

252
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I really have still to
justify the fact that it

253
00:17:09 --> 00:17:12
truly is A transpose.
 

254
00:17:12 --> 00:17:15
The transpose of that
matrix that comes into

255
00:17:15 --> 00:17:17
Kirchhoff's current law.
 

256
00:17:17 --> 00:17:20
I guess in the first minute
of the lecture I looked

257
00:17:20 --> 00:17:21
at that particular node.
 

258
00:17:21 --> 00:17:23
Let's look at all the nodes.
 

259
00:17:23 --> 00:17:28
Let me look at A
transpose w equals here.

260
00:17:28 --> 00:17:28
OK.
 

261
00:17:28 --> 00:17:32
So I'll copy a transpose
because I truly believe

262
00:17:32 --> 00:17:35
that Kirchhoff would
want me to do it.

263
00:17:35 --> 00:17:45
OK, so that becomes a row, that
becomes the next row, of course

264
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I see three guys going into
node one, then the one

265
00:17:51 --> 00:17:53
that I looked at before.
 

266
00:17:53 --> 00:17:59
That's the three edges, node
three, and then the last one.

267
00:17:59 --> 00:18:02
Let's keep the last
one in for now.

268
00:18:02 --> 00:18:08
And because if I didn't fix
that I'd have that last one.

269
00:18:08 --> 00:18:10
There we go, so that
multiplies, what does

270
00:18:10 --> 00:18:14
that multiply now in
Kirchhoff's current law?

271
00:18:14 --> 00:18:16
Multiplies w, so currents.
 

272
00:18:16 --> 00:18:22
So here the currents, one,
two, three, four and five.

273
00:18:22 --> 00:18:26
All I'm doing now is just,
like, convincing you that it

274
00:18:26 --> 00:18:30
really is a transpose, that if
I look at, let me pick that

275
00:18:30 --> 00:18:39
node now, if I look at that
node I see edge 1 coming in, I

276
00:18:39 --> 00:18:43
see edge 3 going out, I see
edge 4 going out and when I

277
00:18:43 --> 00:18:47
look at that second node, I
look here at the second row, I

278
00:18:47 --> 00:18:51
see edge one coming in, edge
three and edge four going out,

279
00:18:51 --> 00:18:55
multiplying those currents, and
that will give, that current

280
00:18:55 --> 00:18:58
balance there gives that zero.
 

281
00:18:58 --> 00:18:59
Right?
 

282
00:18:59 --> 00:19:04
That current balance at node
two gives that zero in

283
00:19:04 --> 00:19:05
the right hand side.
 

284
00:19:05 --> 00:19:09
And then, of course, other
zeroes are here too.

285
00:19:09 --> 00:19:14
This is minus w_1, minus
w_2, so I have a zero.

286
00:19:14 --> 00:19:19
This is with no
current sources.

287
00:19:19 --> 00:19:19
OK.
 

288
00:19:19 --> 00:19:23
And this one, of course.
 

289
00:19:23 --> 00:19:28
I guess I'm hoping that you
say yes, it's the A transpose

290
00:19:28 --> 00:19:34
really was the right
matrix to express in=out,

291
00:19:34 --> 00:19:38
Kirchhoff's current law.
 

292
00:19:38 --> 00:19:41
OK.
 

293
00:19:41 --> 00:19:43
Of course, by now you're
probably getting blase.

294
00:19:43 --> 00:19:45
You expect it to be a
transpose, you don't

295
00:19:45 --> 00:19:47
need any convincing.
 

296
00:19:47 --> 00:19:51
But it's kind of good to see
each time because it's like,

297
00:19:51 --> 00:19:53
well, it's not a miracle.
 

298
00:19:53 --> 00:19:56
But it's like a miracle.
 

299
00:19:56 --> 00:19:57
It's as good as a miracle.
 

300
00:19:57 --> 00:20:02
Because to get a transpose
is just what makes

301
00:20:02 --> 00:20:04
everything right.
 

302
00:20:04 --> 00:20:11
Now, here's a question.
 

303
00:20:11 --> 00:20:15
This is a question
worth thinking about.

304
00:20:15 --> 00:20:17
What are the solutions?
 

305
00:20:17 --> 00:20:22
If I only look at this piece of
the framework, if I just look

306
00:20:22 --> 00:20:26
at Kirchhoff's current law,
it's telling me, and I have

307
00:20:26 --> 00:20:27
zero current sources.
 

308
00:20:27 --> 00:20:31
Well let's take that, let me
take just this piece of the

309
00:20:31 --> 00:20:38
whole framework and ask you
how many vectors, w,

310
00:20:38 --> 00:20:41
how many solutions?
 

311
00:20:41 --> 00:20:42
Are there any solutions?
 

312
00:20:42 --> 00:20:45
Well, of course, there's
always the zero solution.

313
00:20:45 --> 00:20:50
But I always ask you, what are
the solutions when zero is

314
00:20:50 --> 00:20:52
on the right hand side?
 

315
00:20:52 --> 00:20:56
So, what are the w's that solve
Kirchhoff's current law.

316
00:20:56 --> 00:20:58
Now that's a new question.
 

317
00:20:58 --> 00:20:59
We haven't asked that before.
 

318
00:20:59 --> 00:21:04
What we asked before was
the question Au, Au=0,

319
00:21:04 --> 00:21:06
remember it was then A.
 

320
00:21:06 --> 00:21:11
This five by four matrix,
u had four components.

321
00:21:11 --> 00:21:15
Just remind me, and I'm
putting that column back

322
00:21:15 --> 00:21:19
in, so this is still here.
 

323
00:21:19 --> 00:21:23
In the un-reduced,
un-grounded case.

324
00:21:23 --> 00:21:28
Well, just so we get started,
remind me what the solutions

325
00:21:28 --> 00:21:35
u were for Au equals
zero. .

326
00:21:35 --> 00:21:36
Or any multiple of it.
 

327
00:21:36 --> 00:21:38
A whole line of vectors.
 

328
00:21:39 --> 00:21:42
 any constant c.
 

329
00:21:42 --> 00:21:47
OK, so this had, there
were four columns.

330
00:21:47 --> 00:21:49
But only three
were independent.

331
00:21:49 --> 00:21:49
OK.
 

332
00:21:49 --> 00:21:56
Now, now I've made
them into rows.

333
00:21:56 --> 00:21:59
And I made the rows
into columns.

334
00:21:59 --> 00:22:02
So now I have five columns.
 

335
00:22:02 --> 00:22:08
What I'm leading to is, I want
to count in advance how many

336
00:22:08 --> 00:22:12
w's I should be looking for,
and then look for them.

337
00:22:12 --> 00:22:18
OK, so the first question is
how many different solutions w.

338
00:22:18 --> 00:22:22
First of all, are
there some solutions?

339
00:22:22 --> 00:22:25
Is there is a solution
w, other than zero of

340
00:22:25 --> 00:22:27
course, to this system?
 

341
00:22:27 --> 00:22:31
Well, as we said last
time, we've only got

342
00:22:31 --> 00:22:33
four equations here.
 

343
00:22:33 --> 00:22:35
We've got five unknowns.
 

344
00:22:35 --> 00:22:38
Of course there's at
least one solution.

345
00:22:38 --> 00:22:42
Five for equations, five
unknowns, I can do elimination.

346
00:22:42 --> 00:22:45
Whatever systematic procedure
you want me to do.

347
00:22:45 --> 00:22:48
In the end, I'm going
to find a solution.

348
00:22:48 --> 00:22:50
Now, I might find
more solutions.

349
00:22:50 --> 00:22:52
So that's the question.
 

350
00:22:52 --> 00:22:57
So what's the, now this is the
key fact of linear algebra.

351
00:22:57 --> 00:23:02
Which is, it just tells us
the numbers of everything.

352
00:23:02 --> 00:23:07
So this told us that there were
three independent columns.

353
00:23:07 --> 00:23:09
Of A.
 

354
00:23:09 --> 00:23:15
Now, for that key theorem,
which tells me that

355
00:23:15 --> 00:23:19
how many independent
rows of a are there?

356
00:23:19 --> 00:23:19
Three.
 

357
00:23:19 --> 00:23:22
That number is equal.
 

358
00:23:22 --> 00:23:25
I just want to say that's
a pretty remarkable fact.

359
00:23:25 --> 00:23:34
If I have a 50 by 80 matrix and
that 50 by 80 matrix has 17

360
00:23:34 --> 00:23:39
independent columns, then this
great fact tells me that there

361
00:23:39 --> 00:23:41
are 17 independent rows.
 

362
00:23:41 --> 00:23:49
And if those 50 times 80, 4,000
numbers are random, boy, you

363
00:23:49 --> 00:23:52
can't look at it and
see what are they.

364
00:23:52 --> 00:23:53
Independent rows.
 

365
00:23:53 --> 00:23:56
But if you know there are 17
independent columns then there

366
00:23:56 --> 00:23:58
are 17 independent rows.
 

367
00:23:58 --> 00:24:01
So, what does that mean here?
 

368
00:24:01 --> 00:24:07
That means that out of the five
columns of A transpose, which

369
00:24:07 --> 00:24:10
were rows of A, three
are independent.

370
00:24:10 --> 00:24:14
So just tell me, how many
solutions I'm looking for.

371
00:24:14 --> 00:24:16
Before I look for them.
 

372
00:24:16 --> 00:24:21
So the number of the number of
independent w's, independent

373
00:24:21 --> 00:24:25
solutions will be what?
 

374
00:24:25 --> 00:24:27
What's your guess?
 

375
00:24:27 --> 00:24:28
Two!
 

376
00:24:28 --> 00:24:30
Two is the right guess.
 

377
00:24:30 --> 00:24:34
Two, because I have all
together five unknowns I

378
00:24:34 --> 00:24:40
subtract three equations that
are really there, I have three

379
00:24:40 --> 00:24:43
real equations there even
though it looks like four.

380
00:24:43 --> 00:24:44
And I get two.
 

381
00:24:44 --> 00:24:50
So that general picture is n -
oh, I'm sorry it's actually

382
00:24:50 --> 00:24:54
m because I'm doing
the transpose here.

383
00:24:54 --> 00:25:00
So it's m w's minus
r, the rank.

384
00:25:00 --> 00:25:05
So that's m is five, the rank
is three and this counts the

385
00:25:05 --> 00:25:07
number of independent
solutions.

386
00:25:07 --> 00:25:12
So it's a nice, it
couldn't be better.

387
00:25:12 --> 00:25:17
It's a fundamental count of
how many solutions are there.

388
00:25:17 --> 00:25:22
You really taking the number of
unknowns, five, and you're

389
00:25:22 --> 00:25:25
subtracting the number of
equations that are really

390
00:25:25 --> 00:25:28
there, three and that leaves
you with two solutions which we

391
00:25:28 --> 00:25:30
will have to find in a minute.
 

392
00:25:30 --> 00:25:35
Can you say why there's really
only three equations there?

393
00:25:35 --> 00:25:41
Why do I say that that
fourth equation is not

394
00:25:41 --> 00:25:45
contributing anything new?
 

395
00:25:45 --> 00:25:50
I believe that that fourth
equation is a consequence

396
00:25:50 --> 00:25:52
of the first three.
 

397
00:25:52 --> 00:25:56
And, therefore, if it's there
or if it's not there, it's not

398
00:25:56 --> 00:25:58
telling me anything
new about in=out.

399
00:25:59 --> 00:26:03
In other words, if I have a
closed system, closed because

400
00:26:03 --> 00:26:07
it's zero on the right side, if
I have a closed system and I

401
00:26:07 --> 00:26:12
have in=out at those three
nodes, then I'll automatically

402
00:26:12 --> 00:26:19
have in=out at the fourth node,
because the total in is zero.

403
00:26:19 --> 00:26:20
And the total out is zero.
 

404
00:26:20 --> 00:26:23
So if I'm right at three I'll
be right at the fourth one.

405
00:26:23 --> 00:26:27
And now just tell me with the
numbers, how would I get

406
00:26:27 --> 00:26:37
this equation w_4+w_5=0
from the first three?

407
00:26:37 --> 00:26:40
Well, it's probably the same
way that I got that column,

408
00:26:40 --> 00:26:44
that that column was
connected to those columns.

409
00:26:44 --> 00:26:46
What do I do?
 

410
00:26:46 --> 00:26:48
Add them.
 

411
00:26:48 --> 00:26:52
If you add that equation to
that equation to that equation,

412
00:26:52 --> 00:26:56
add those three equations, what
happens? the w_1's cancel, the

413
00:26:56 --> 00:27:01
w_2's cancel, the w_3's cancel,
this says there's a minus w_4

414
00:27:01 --> 00:27:06
and a minus w_5 that adds to
zero plus zero plus zero, minus

415
00:27:06 --> 00:27:11
w_4, minus w_5 equalling zero
is the same as plus w_4

416
00:27:11 --> 00:27:14
plus w_5 equals zero.
 

417
00:27:14 --> 00:27:18
The four equations add
to zero equals zero.

418
00:27:18 --> 00:27:21
That's the central thing is.
 

419
00:27:21 --> 00:27:25
The four equations add to zero
equals zero just the way the

420
00:27:25 --> 00:27:28
four columns up here added
to the zero column.

421
00:27:28 --> 00:27:29
OK.
 

422
00:27:29 --> 00:27:32
So we got the count.
 

423
00:27:32 --> 00:27:37
Now, this is the
interesting part, always.

424
00:27:37 --> 00:27:38
What are the solutions?
 

425
00:27:38 --> 00:27:40
What are the actual w's?
 

426
00:27:40 --> 00:27:43
OK.
 

427
00:27:43 --> 00:27:49
You could say, wait a minute,
you're asking me to solve four

428
00:27:49 --> 00:27:53
equations and five unknowns,
and just say what

429
00:27:53 --> 00:27:53
the answer is.
 

430
00:27:53 --> 00:27:57
Well, normally that's
not reasonable.

431
00:27:57 --> 00:28:02
But here we can get
help from the graph.

432
00:28:02 --> 00:28:04
Let me give you an idea here.
 

433
00:28:04 --> 00:28:08
So what are we looking
for on the graph?

434
00:28:08 --> 00:28:09
That solves it.
 

435
00:28:09 --> 00:28:13
We're looking for a
bunch of currents that

436
00:28:13 --> 00:28:15
balance themselves.
 

437
00:28:15 --> 00:28:15
Right?
 

438
00:28:15 --> 00:28:17
We've got zero on the
right hand side.

439
00:28:17 --> 00:28:20
So we're not getting
any help from outside.

440
00:28:20 --> 00:28:25
How could you send current
in this loop in a way that

441
00:28:25 --> 00:28:27
would satisfy Kirchhoff.
 

442
00:28:27 --> 00:28:29
He'd be happy.
 

443
00:28:29 --> 00:28:32
The balance load would be true.
 

444
00:28:32 --> 00:28:37
OK. b, I'm just
looking at currents.

445
00:28:37 --> 00:28:41
Current w_1, w_2, w_3.
 

446
00:28:41 --> 00:28:46
Is there any combination of
w_1, w_2, w_3 that would

447
00:28:46 --> 00:28:52
balance itself, that
would make Kirchhoff OK?

448
00:28:52 --> 00:28:55
Well, here's the idea.
 

449
00:28:55 --> 00:28:58
Send that current
around a loop.

450
00:28:58 --> 00:29:05
Loop currents are solutions
to Kirchhoff's balance law.

451
00:29:05 --> 00:29:10
If I send an amp on that edge,
on that edge and backward

452
00:29:10 --> 00:29:13
on that edge, right?
 

453
00:29:13 --> 00:29:17
It's around a loop at every
node, it's totally OK.

454
00:29:17 --> 00:29:24
So I believe that a particular
solution will be for these

455
00:29:24 --> 00:29:30
things to be, let's see what
did I say? w_1, I'll send one

456
00:29:30 --> 00:29:35
around. w_3 will be a one. w_2
was backwards, it wasn't

457
00:29:35 --> 00:29:37
traveling on w_4.
 

458
00:29:37 --> 00:29:39
I think that's a solution.
 

459
00:29:39 --> 00:29:43
That loop current gives me a
solution so let me call this

460
00:29:43 --> 00:29:49
solution, that's the
first solution.

461
00:29:49 --> 00:29:51
And if we do these
multiplications, of course

462
00:29:51 --> 00:29:53
it's going to come out right.
 

463
00:29:53 --> 00:29:58
OK, so that's
solution number one.

464
00:29:58 --> 00:30:01
A w that works.
 

465
00:30:01 --> 00:30:04
OK, with that hint, tell
me a second w that works.

466
00:30:04 --> 00:30:08
In fact, since there are only
two, you'll be giving me

467
00:30:08 --> 00:30:12
the rest of the answer.
 

468
00:30:12 --> 00:30:17
So that was a loop current
that went around that loop.

469
00:30:17 --> 00:30:20
Tell me another one.
 

470
00:30:20 --> 00:30:25
Well, we're a big class
but everybody's seen it.

471
00:30:25 --> 00:30:30
How about around that loop?
 

472
00:30:30 --> 00:30:35
So that would be another thing
and it's pretty clearly

473
00:30:35 --> 00:30:37
not the same as this one.
 

474
00:30:37 --> 00:30:41
So I'm really truly finding
two independent solutions.

475
00:30:41 --> 00:30:43
And what is that
second solution?

476
00:30:43 --> 00:30:48
Let me maybe just
put it in here.

477
00:30:48 --> 00:30:52
And they're both giving
me .

478
00:30:52 --> 00:30:56
And now, what is this
number two solution

479
00:30:56 --> 00:30:57
the loop number two?
 

480
00:30:57 --> 00:31:00
One and two are
not involved now.

481
00:31:00 --> 00:31:07
Number three, see I'm usually
sending it counterclockwise,

482
00:31:07 --> 00:31:09
that's the sort of convention.
 

483
00:31:09 --> 00:31:14
But you know, of course you
just have to follow some

484
00:31:14 --> 00:31:17
convention in connection
with the arrows.

485
00:31:17 --> 00:31:20
So that would go backwards
on three, I think.

486
00:31:20 --> 00:31:24
Forwards on four, and
backwards on five.

487
00:31:24 --> 00:31:27
So that would be
solution number two.

488
00:31:27 --> 00:31:29
OK.
 

489
00:31:29 --> 00:31:32
Now, tell me what all
the solutions are.

490
00:31:32 --> 00:31:36
I found two particular
solutions, two particular loop

491
00:31:36 --> 00:31:39
currents, particularly easy.
 

492
00:31:39 --> 00:31:45
What would be all the solutions
to Kirchhoff's current law,

493
00:31:45 --> 00:31:47
A transpose w equals zero?
 

494
00:31:47 --> 00:31:52
Every w now since I've found
the right number, every w will

495
00:31:52 --> 00:31:57
be a combination of those two.
 

496
00:31:57 --> 00:31:59
Ah, well wait a minute.
 

497
00:31:59 --> 00:32:01
Have I got them all?
 

498
00:32:01 --> 00:32:04
I should have thought,
what about current

499
00:32:04 --> 00:32:07
around the big loop?
 

500
00:32:07 --> 00:32:10
That would certainly
satisfy Kirchhoff.

501
00:32:10 --> 00:32:19
Plus one, one, so why is
this not number three?

502
00:32:19 --> 00:32:23
Around the big loop I have
a one and then a one on

503
00:32:23 --> 00:32:24
the fourth position.
 

504
00:32:24 --> 00:32:28
Backwards on five,
backwards on two.

505
00:32:28 --> 00:32:32
And so there is number,
so I'll put number three

506
00:32:32 --> 00:32:36
with a question mark.
 

507
00:32:36 --> 00:32:41
What's up with that guy?
 

508
00:32:41 --> 00:32:45
It, you know if unless
mathematics has got to close

509
00:32:45 --> 00:32:51
up shop, this better be
a combination of those.

510
00:32:51 --> 00:32:52
And of course, it is.
 

511
00:32:52 --> 00:32:56
If I send something around the
top loop and something around

512
00:32:56 --> 00:33:00
the second loop and add them
together, they'll cancel on the

513
00:33:00 --> 00:33:04
middle edge there and
produce number three.

514
00:33:04 --> 00:33:07
So this is probably
just the sum.

515
00:33:07 --> 00:33:11
If I add that one to that one,
I think I get number three.

516
00:33:11 --> 00:33:14
So number three is true.
 

517
00:33:14 --> 00:33:17
It's a solution but
it's not a new one.

518
00:33:17 --> 00:33:20
OK.
 

519
00:33:20 --> 00:33:25
That was simple, right, once we
saw that loops gave the answer.

520
00:33:25 --> 00:33:33
But, it's, actually it's
quite important and

521
00:33:33 --> 00:33:35
appears everywhere.
 

522
00:33:35 --> 00:33:42
In fact the theory of
electrical networks,

523
00:33:42 --> 00:33:43
current laws and so on.
 

524
00:33:43 --> 00:33:46
I mean that used to be a,
like, basic course in

525
00:33:46 --> 00:33:47
electrical engineering.
 

526
00:33:47 --> 00:33:52
There was a text by Professor
Ernst Guillemin, I remember.

527
00:33:52 --> 00:33:56
It's sort of not so central
to the world any more.

528
00:33:56 --> 00:34:03
And now, but the structure
is just right somehow.

529
00:34:03 --> 00:34:08
And what I wanted to say is you
could take, in those days you

530
00:34:08 --> 00:34:11
maybe took loop currents
as the unknowns.

531
00:34:11 --> 00:34:14
You could think of currents
in the loops as your

532
00:34:14 --> 00:34:16
principal unknowns.
 

533
00:34:16 --> 00:34:17
We don't do that now.
 

534
00:34:17 --> 00:34:23
But, oh, there's, yeah
it comes up again.

535
00:34:23 --> 00:34:27
Knowing all the solutions to A
transpose w equals zero, well,

536
00:34:27 --> 00:34:31
you'll see, what's ahead?
 

537
00:34:31 --> 00:34:35
It will be the continuous
analog of this, where I have

538
00:34:35 --> 00:34:40
flows, not just around a
graph, but in a region.

539
00:34:40 --> 00:34:44
And Laplace's equation is going
to come up, and the equations

540
00:34:44 --> 00:34:49
of divergence and gradient,
all that great stuff is

541
00:34:49 --> 00:34:50
coming in Chapter 3.
 

542
00:34:50 --> 00:34:57
And this is somehow
the discrete case.

543
00:34:57 --> 00:35:00
If by these these loop
currents that has something

544
00:35:00 --> 00:35:03
to do with the curl.
 

545
00:35:03 --> 00:35:08
And these differences that
a takes has something

546
00:35:08 --> 00:35:09
to do with gradients.
 

547
00:35:09 --> 00:35:13
And this, Kirchhoff's current
law has something to

548
00:35:13 --> 00:35:15
do with divergence.
 

549
00:35:15 --> 00:35:21
Can I just say ahead of time,
what we're doing is really good

550
00:35:21 --> 00:35:28
to see and get it because then
you have a way to understand

551
00:35:28 --> 00:35:33
vector calculus.
 

552
00:35:33 --> 00:35:36
This is discrete vector
calculus we're doing.

553
00:35:36 --> 00:35:37
OK.
 

554
00:35:37 --> 00:35:40
Now, it's just right.
 

555
00:35:40 --> 00:35:45
Forgive me for my sermon here.
 

556
00:35:45 --> 00:35:45
Alright.
 

557
00:35:45 --> 00:35:50
Now, may I bring the
pieces together finally?

558
00:35:50 --> 00:35:58
May I bring the three steps
together into, well first

559
00:35:58 --> 00:36:01
you would say bring
them into one equation.

560
00:36:01 --> 00:36:05
Put the three steps, combine
the three steps into one.

561
00:36:05 --> 00:36:06
OK.
 

562
00:36:06 --> 00:36:10
So, what happens if I do that?
 

563
00:36:10 --> 00:36:18
I take that last step A
transpose f, is A transpose w.

564
00:36:18 --> 00:36:24
So now I'm going to
get one equation.

565
00:36:24 --> 00:36:27
Which is the equation that's
going to have the stiffness

566
00:36:27 --> 00:36:30
matrix in it, A transpose C A.
 

567
00:36:30 --> 00:36:35
It's the conductance matrix.
 

568
00:36:35 --> 00:36:40
And it's the equation that a
big finite element code, a

569
00:36:40 --> 00:36:42
circuit simulation code.
 

570
00:36:42 --> 00:36:45
It's the matrix they have
to find and work with.

571
00:36:45 --> 00:36:50
And those codes are enormous,
and spice by the way is

572
00:36:50 --> 00:36:54
the sort of grandfather of
circuit simulation code.

573
00:36:54 --> 00:36:58
Somebody at Berkeley had the
sense to see hey, we've

574
00:36:58 --> 00:37:00
got giant circuits now.
 

575
00:37:00 --> 00:37:03
Modern circuits have
thousands of elements.

576
00:37:03 --> 00:37:09
And you can't do it by eye the
way we can do this one by eye.

577
00:37:09 --> 00:37:13
You've got to organize it
and write a code, and

578
00:37:13 --> 00:37:14
spice is the start.
 

579
00:37:14 --> 00:37:18
So one way to do it is to
end up with one equation.

580
00:37:18 --> 00:37:22
So that was f equals
A transpose w.

581
00:37:22 --> 00:37:25
I'm now going, I'm just
assembling the whole loop.

582
00:37:25 --> 00:37:27
But w is Ce.
 

583
00:37:28 --> 00:37:41
So that's A transpose Ce,
but e is b minus Au.

584
00:37:41 --> 00:37:43
Nothing new there.
 

585
00:37:43 --> 00:37:48
Nothing new maybe except that
it involves both the f and the

586
00:37:48 --> 00:37:54
b, where our earlier examples
involved either an f, in masses

587
00:37:54 --> 00:37:56
and springs, or a
b in the squares.

588
00:37:56 --> 00:37:58
Now they're both here.
 

589
00:37:58 --> 00:38:01
So now, that's my
equation, f equals this.

590
00:38:01 --> 00:38:06
And now let me just move that
to the, let me just recollect

591
00:38:06 --> 00:38:10
it, that's A transpose C A, the
big thing that I wanted to see.

592
00:38:10 --> 00:38:13
I'll put it on the left side.
 

593
00:38:13 --> 00:38:15
And what will I have
on the right side?

594
00:38:15 --> 00:38:21
I'll have A transpose C b.
 

595
00:38:21 --> 00:38:24
And I'll have f is now
coming over to the other

596
00:38:24 --> 00:38:28
side with a minus x.
 

597
00:38:28 --> 00:38:31
That's the big equation.
 

598
00:38:31 --> 00:38:33
You could say that's the
fundamental equation

599
00:38:33 --> 00:38:35
of equilibrium.
 

600
00:38:35 --> 00:38:38
And you see how it's right?
 

601
00:38:38 --> 00:38:43
It involves the A transpose
C A, which we expect.

602
00:38:43 --> 00:38:45
Over here was the
A transpose C b.

603
00:38:45 --> 00:38:52
Now, which problem, before
networks, produced an A

604
00:38:52 --> 00:38:56
transpose b or an
A transpose C b?

605
00:38:56 --> 00:38:57
That was least squared.
 

606
00:38:57 --> 00:39:01
And now, so that's the
least squares, the b term.

607
00:39:01 --> 00:39:05
The b is there with a couple of
matrices because b entered the

608
00:39:05 --> 00:39:07
problem just one step around.
 

609
00:39:07 --> 00:39:09
It had two more steps to go.
 

610
00:39:09 --> 00:39:13
It had a C step and then
an A transpose step.

611
00:39:13 --> 00:39:16
Whereas f up here is at the
very end and now it appears

612
00:39:16 --> 00:39:19
with a minus sign.
 

613
00:39:19 --> 00:39:24
That's different from springs
and masses simply because, the

614
00:39:24 --> 00:39:26
sign conventions,
you could say.

615
00:39:26 --> 00:39:28
OK, there is the equation.
 

616
00:39:28 --> 00:39:30
OK, fine.
 

617
00:39:30 --> 00:39:34
So that's what you
have to solve.

618
00:39:34 --> 00:39:36
And actually I think of that
as the fundamental problem

619
00:39:36 --> 00:39:38
of numerical analysis.
 

620
00:39:38 --> 00:39:40
How to solve that equation.
 

621
00:39:40 --> 00:39:44
More effort, more thinking
goes into that than probably

622
00:39:44 --> 00:39:46
any other single problem.
 

623
00:39:46 --> 00:39:48
In some form.
 

624
00:39:48 --> 00:39:52
OK, and here's some
part of that thinking.

625
00:39:52 --> 00:39:55
Part of that thinking and
another important possibility

626
00:39:55 --> 00:40:00
is to keep, this was like the
one equation, the one field

627
00:40:00 --> 00:40:06
problem, this corresponds to
the displacement method.

628
00:40:06 --> 00:40:12
Can I use words that I'm not
going to use seriously for

629
00:40:12 --> 00:40:19
another few weeks, this would
correspond to the displacement

630
00:40:19 --> 00:40:22
method in finite elements.
 

631
00:40:22 --> 00:40:25
In f em, f em for
finite element method.

632
00:40:25 --> 00:40:27
That's the displacement
method, it's the method

633
00:40:27 --> 00:40:30
that that most people use.
 

634
00:40:30 --> 00:40:33
It's the standard method.
 

635
00:40:33 --> 00:40:33
OK.
 

636
00:40:33 --> 00:40:37
But it's the only possibility,
and let me show you a second

637
00:40:37 --> 00:40:40
one that involves
two equations.

638
00:40:40 --> 00:40:45
Because that's also very
important. with many

639
00:40:45 --> 00:40:46
other applications.
 

640
00:40:46 --> 00:40:49
That we will see but
haven't seen yet.

641
00:40:49 --> 00:40:54
So my two equations, I really
should say two systems,

642
00:40:54 --> 00:40:58
because one equation,
that's a vector of course.

643
00:40:58 --> 00:41:01
So I have a system of two
vector equations, it would

644
00:41:01 --> 00:41:04
go into a block matrix
and you'll see it.

645
00:41:04 --> 00:41:11
OK, so what what two unknowns
am I going to keep?

646
00:41:11 --> 00:41:12
u, I'm keeping.
 

647
00:41:12 --> 00:41:14
Displacement, I'm keeping.
 

648
00:41:14 --> 00:41:17
But I'm also going to keep
what I think of as the

649
00:41:17 --> 00:41:20
other important unknown, w.
 

650
00:41:20 --> 00:41:22
So the other important
unknown is w.

651
00:41:22 --> 00:41:26
So now I have two equations and
one of them is just that, is

652
00:41:26 --> 00:41:30
just the current, is just the
current law, the balance

653
00:41:30 --> 00:41:35
law A transpose w equal f.
 

654
00:41:35 --> 00:41:39
The only guy I'm
eliminating is e.

655
00:41:39 --> 00:41:45
Initially you could say I've a
three field system. u, e and w.

656
00:41:45 --> 00:41:50
Now, e and w are so easily
connected that I'm going to

657
00:41:50 --> 00:41:53
eliminate e. e is c inverse w.
 

658
00:41:53 --> 00:41:56
I did it actually here.
 

659
00:41:56 --> 00:42:03
This w is Cb-Au, that's we know
if I multiply by the C inverse,

660
00:42:03 --> 00:42:08
then I have C inverse w is, and
I bring the Au over to the

661
00:42:08 --> 00:42:12
far left and I have
only the b left.

662
00:42:12 --> 00:42:21
Everybody saw that, I did a C
inverse there to get e off by

663
00:42:21 --> 00:42:27
itself and then I substituted
for e, I put in the u part so

664
00:42:27 --> 00:42:32
e's now gone, and the
equation is C inverse

665
00:42:32 --> 00:42:40
w plus Au equals b.
 

666
00:42:40 --> 00:42:46
That's my two field system.
 

667
00:42:46 --> 00:42:50
Now, there's a matrix here.
 

668
00:42:50 --> 00:42:53
This is really nice.
 

669
00:42:53 --> 00:42:54
I just want to write that.
 

670
00:42:54 --> 00:42:57
I've got to what I want but
now I want to look at it.

671
00:42:57 --> 00:43:01
So I think of a two by two
block matrix multiplying

672
00:43:01 --> 00:43:06
wu and giving me bf.
 

673
00:43:07 --> 00:43:10
And I want to ask you
about that matrix.

674
00:43:10 --> 00:43:18
So this is the matrix for when
I've only eliminated e and I've

675
00:43:18 --> 00:43:21
still got w as well as u.
 

676
00:43:21 --> 00:43:26
OK, you can read off
what's in that matrix.

677
00:43:26 --> 00:43:28
What goes up here?
 

678
00:43:28 --> 00:43:30
C inverse, of course.
 

679
00:43:30 --> 00:43:33
Positive diagonal, usually.
 

680
00:43:33 --> 00:43:34
Easy.
 

681
00:43:34 --> 00:43:36
What goes here is a
rectangular, that

682
00:43:36 --> 00:43:37
guy is rectangular.
 

683
00:43:37 --> 00:43:43
A transpose is multiplying w,
so it goes down here and this

684
00:43:43 --> 00:43:47
equation has no u in it.
 

685
00:43:47 --> 00:43:55
That matrix is worth noticing.
 

686
00:43:55 --> 00:43:58
And let's spend the rest of
this, the remaining minutes

687
00:43:58 --> 00:44:00
just to think about
that matrix.

688
00:44:00 --> 00:44:06
I just want to say what if
I keep w and u, this is

689
00:44:06 --> 00:44:08
an important possibility.
 

690
00:44:08 --> 00:44:13
And it's important in finite
elements which as you know is

691
00:44:13 --> 00:44:16
just a terrific way to
solve a whole lot of

692
00:44:16 --> 00:44:18
continuum problems.
 

693
00:44:18 --> 00:44:23
And what's it called when I
have w and u together, both

694
00:44:23 --> 00:44:27
unknowns, not eliminating
w now, it's called

695
00:44:27 --> 00:44:29
the mixed method.
 

696
00:44:29 --> 00:44:33
So this corresponds to
the mixed method in

697
00:44:33 --> 00:44:36
finite elements.
 

698
00:44:36 --> 00:44:45
It corresponds to the
possibility of keeping w and u.

699
00:44:45 --> 00:44:47
Well, and you might say,
wait, isn't there a

700
00:44:47 --> 00:44:50
third possibility?
 

701
00:44:50 --> 00:44:52
And what would that be?
 

702
00:44:52 --> 00:44:56
Keep only w.
 

703
00:44:56 --> 00:45:00
Here we kept only u, here
we've got them both, this is

704
00:45:00 --> 00:45:07
kind of the mother of all
equilibrium equations.

705
00:45:07 --> 00:45:11
And another possibly would
be to keep only the w's,

706
00:45:11 --> 00:45:14
to make the currents
the primary unknowns.

707
00:45:14 --> 00:45:20
And that in the finite element
structural context, that would

708
00:45:20 --> 00:45:23
be saying make the stresses.
 

709
00:45:23 --> 00:45:26
So of course it'd be
called the stress method.

710
00:45:26 --> 00:45:28
It'd be called the stress
method, and Professor Pian in

711
00:45:28 --> 00:45:33
Course 16, now retired, was
one of the major developers

712
00:45:33 --> 00:45:35
of the stress method.
 

713
00:45:35 --> 00:45:39
The difficulty with the stress
method, the reason it didn't

714
00:45:39 --> 00:45:45
win big time, is that the w's,
if you make them the

715
00:45:45 --> 00:45:47
unknowns you've got a
constraint on them.

716
00:45:47 --> 00:45:52
Kirchhoff's, not all
w's are allowed.

717
00:45:52 --> 00:45:57
Somehow over here all u's are
allowed, and that made it much

718
00:45:57 --> 00:46:00
easier to set up the problem.
 

719
00:46:00 --> 00:46:05
So the displacement method
is the 95 percent winner.

720
00:46:05 --> 00:46:10
But there are problems where
maybe C inverse is complicated,

721
00:46:10 --> 00:46:17
or C is too complicated
and you're better to.

722
00:46:17 --> 00:46:18
We can see that.
 

723
00:46:18 --> 00:46:22
That's later in the book,
but we want to see now

724
00:46:22 --> 00:46:24
about that matrix.
 

725
00:46:24 --> 00:46:29
Well, if I wrote that matrix
down, and let me write just

726
00:46:29 --> 00:46:34
so you, I want to ask you
about that block matrix.

727
00:46:34 --> 00:46:36
What's its size.
 

728
00:46:36 --> 00:46:40
Now just focus entirely on
that block matrix, because

729
00:46:40 --> 00:46:41
that's what I care about.
 

730
00:46:41 --> 00:46:46
What's the size of that matrix?
 

731
00:46:46 --> 00:46:50
Let's see, what's the
size of C? m by m.

732
00:46:50 --> 00:46:53
What's the size of A? n by n.
 

733
00:46:53 --> 00:46:54
So what do I have here?
 

734
00:46:54 --> 00:46:58
I've got n rows and m
plus n columns, and

735
00:46:58 --> 00:46:59
there's n more rows.
 

736
00:46:59 --> 00:47:01
It's of size m+n.
 

737
00:47:01 --> 00:47:05
 
 

738
00:47:05 --> 00:47:08
It's got the n u's
and the n w's.

739
00:47:08 --> 00:47:10
Of course, m+n.
 

740
00:47:10 --> 00:47:12
And the natural size, right.
 

741
00:47:12 --> 00:47:17
So it's got more unknowns but
we'll see, oh in optimization

742
00:47:17 --> 00:47:19
you bring in Lagrange
multipliers, that's just

743
00:47:19 --> 00:47:23
exactly parallel to
what we're doing here.

744
00:47:23 --> 00:47:26
You have more, you have
extra bunch of unknowns.

745
00:47:26 --> 00:47:27
That's what we have.
 

746
00:47:27 --> 00:47:32
Now what else about
that matrix?

747
00:47:32 --> 00:47:35
I was going to write
down a very, very tiny

748
00:47:35 --> 00:47:37
model for that matrix.
 

749
00:47:37 --> 00:47:39
I'll just make it two by two.
 

750
00:47:39 --> 00:47:44
Here's a model for that
matrix where C is just a

751
00:47:44 --> 00:47:48
one and A is just a one.
 

752
00:47:48 --> 00:47:51
I mean, it's kind of
laughable, right?

753
00:47:51 --> 00:47:53
That model, this is
the real thing.

754
00:47:53 --> 00:48:00
But it gives you an
example to check against.

755
00:48:00 --> 00:48:04
OK, what's a property
of that matrix?

756
00:48:04 --> 00:48:06
It's, again?
 

757
00:48:06 --> 00:48:06
Symmetric.
 

758
00:48:06 --> 00:48:07
Good.
 

759
00:48:07 --> 00:48:09
That's a symmetric matrix.
 

760
00:48:09 --> 00:48:13
Because what happens if I
transpose that block matrix?

761
00:48:13 --> 00:48:19
That A block will flip over
here as A transpose, the A

762
00:48:19 --> 00:48:23
transpose block will flip up
there as A, what happens

763
00:48:23 --> 00:48:26
to the C inverse block?
 

764
00:48:26 --> 00:48:28
C is a symmetric guide.
 

765
00:48:28 --> 00:48:32
In fact, it was just diagonal
in our imagination.

766
00:48:32 --> 00:48:37
The key point it's symmetric,
its inverse is symmetric,

767
00:48:37 --> 00:48:39
its transpose is the same.
 

768
00:48:39 --> 00:48:42
So that's symmetric matrix.
 

769
00:48:42 --> 00:48:43
That's a good thing.
 

770
00:48:43 --> 00:48:46
Right, now we've got a
matrix that's symmetric,

771
00:48:46 --> 00:48:49
square symmetric.
 

772
00:48:49 --> 00:48:53
OK, what's my other question
about that matrix?

773
00:48:53 --> 00:49:00
Is it or is it not
positive definite, right?

774
00:49:00 --> 00:49:03
We've got to answer
that question.

775
00:49:03 --> 00:49:06
Have we got a positive
definite matrix?

776
00:49:06 --> 00:49:08
Would all the pivots
be positive.

777
00:49:08 --> 00:49:11
Would the eigenvalues
be positive?

778
00:49:11 --> 00:49:15
What's your guess?
 

779
00:49:15 --> 00:49:16
No.
 

780
00:49:16 --> 00:49:19
That matrix is not positive.
 

781
00:49:19 --> 00:49:24
No way that a matrix with a
zero there, a zero block, or

782
00:49:24 --> 00:49:27
that matrix with a zero number
could be positive definite.

783
00:49:27 --> 00:49:28
No, no way.
 

784
00:49:28 --> 00:49:39
The energy in this guy, this u
transpose Au, you remember,

785
00:49:39 --> 00:49:46
would be u_1 squared
and u_1*u_2, twice.

786
00:49:46 --> 00:49:50
And no u_2 squareds.
 

787
00:49:50 --> 00:49:55
And that thing is
definitely indefinite.

788
00:49:55 --> 00:49:59
Right?
 

789
00:49:59 --> 00:50:02
In the u_1 direction
it looks good, that's

790
00:50:02 --> 00:50:03
things positive there.
 

791
00:50:03 --> 00:50:09
But if I took u_1 and u_2 to
have opposite signs, and made

792
00:50:09 --> 00:50:12
u_2 big enough then of course
this just brings it down.

793
00:50:12 --> 00:50:16
So the graph of that
would be a saddle point.

794
00:50:16 --> 00:50:18
The graph of that
would be a saddle.

795
00:50:18 --> 00:50:26
OK, and now here I have the
same thing on an m+n size.

796
00:50:26 --> 00:50:26
So what do I have?
 

797
00:50:26 --> 00:50:28
Actually, you could see.
 

798
00:50:28 --> 00:50:34
The last exercise is mentally
do elimination on that matrix.

799
00:50:34 --> 00:50:37
Mentally do elimination
on that matrix.

800
00:50:37 --> 00:50:42
So start with the first m rows.
 

801
00:50:42 --> 00:50:45
We'll work with those first.
 

802
00:50:45 --> 00:50:48
What will elimination do, what
will the pivots be like,

803
00:50:48 --> 00:50:51
what will happen at the
beginning of elimination?

804
00:50:51 --> 00:50:55
When I start with this matrix?
 

805
00:50:55 --> 00:50:59
Well it meets C inverse right
away, that diagonal matrix,

806
00:50:59 --> 00:51:00
and it's extremely happy.
 

807
00:51:00 --> 00:51:02
Those will be the
pivots, right?

808
00:51:02 --> 00:51:04
They're sitting on
the diagonals, zero

809
00:51:04 --> 00:51:05
off the diagonals.
 

810
00:51:05 --> 00:51:08
They'll be positive pivots,
I'll have m positive

811
00:51:08 --> 00:51:09
pivots here.
 

812
00:51:09 --> 00:51:15
And then I get down to where
a comes in the picture.

813
00:51:15 --> 00:51:21
So on the last board here, let
me just copy this matrix, [C

814
00:51:21 --> 00:51:27
inverse, A; A transpose, 0], an
elimination is going to, it'll

815
00:51:27 --> 00:51:29
be very happy with that.
 

816
00:51:29 --> 00:51:35
But it's going to put, so
it's happy with that row.

817
00:51:35 --> 00:51:36
Block row.
 

818
00:51:36 --> 00:51:40
It's going to do an elimination
to get a bunch of zeroes

819
00:51:40 --> 00:51:45
there and what did it do?
 

820
00:51:45 --> 00:51:48
This was elimination, this
was subtracting, yeah what

821
00:51:48 --> 00:51:50
did it subtract here?
 

822
00:51:50 --> 00:51:55
It multiplied these pivot rows
by something and subtracted

823
00:51:55 --> 00:52:01
from these lower rows
and got the zero block.

824
00:52:01 --> 00:52:04
And what did it multiply by?
 

825
00:52:04 --> 00:52:08
What do I multiply that block
row, and this is a perfect,

826
00:52:08 --> 00:52:12
perfect exercise to see how
blocks are just like numbers.

827
00:52:12 --> 00:52:14
You can deal with them.
 

828
00:52:14 --> 00:52:18
What do I multiply that block
row by and subtract from the

829
00:52:18 --> 00:52:22
row below to get a zero.
 

830
00:52:22 --> 00:52:27
You said C A transpose, but
I don't think that's it.

831
00:52:27 --> 00:52:28
A transpose C.
 

832
00:52:28 --> 00:52:31
You've got to multiply
by A transpose C.

833
00:52:31 --> 00:52:34
First of all, C A transpose
wouldn't be a possibility.

834
00:52:34 --> 00:52:38
Wrong shapes. a transpose
those C is the four by

835
00:52:38 --> 00:52:42
five, five by five guy.
 

836
00:52:42 --> 00:52:46
So you multiply A transpose C,
that cancels that, leaves the a

837
00:52:46 --> 00:52:49
transpose, when you subtract it
gives you a zero, and what

838
00:52:49 --> 00:52:51
does it give you there?
 

839
00:52:51 --> 00:52:55
What shows up there?
 

840
00:52:55 --> 00:53:01
A transpose C multiplies at A,
subtracts so it's actually

841
00:53:01 --> 00:53:06
what shows up there is
minus A transpose C A.

842
00:53:06 --> 00:53:08
Let me write it in there.
 

843
00:53:08 --> 00:53:10
Minus A transpose C A.
 

844
00:53:10 --> 00:53:13
So that matrix is exactly
what comes from this one.

845
00:53:13 --> 00:53:16
It's exactly what we do
when we eliminate w.

846
00:53:16 --> 00:53:18
That's what elimination is.
 

847
00:53:18 --> 00:53:21
I just eliminated w by
getting a zero there.

848
00:53:21 --> 00:53:25
And I got only an equation,
but notice the minus.

849
00:53:25 --> 00:53:32
So, final question, what are
the signs of the last n pivots?

850
00:53:32 --> 00:53:35
The first m were all positive,
and they were sitting on

851
00:53:35 --> 00:53:38
the diagonal already.
 

852
00:53:38 --> 00:53:43
The last n are not so easy
to see, but we can see

853
00:53:43 --> 00:53:44
what sign they have.
 

854
00:53:44 --> 00:53:48
And what sign do the
last n pivots have?

855
00:53:48 --> 00:53:50
Minus.
 

856
00:53:50 --> 00:53:53
Because they come from
a negative definite.

857
00:53:53 --> 00:53:55
Minus A transpose C A
is shown up there.

858
00:53:55 --> 00:53:58
So that's the saddle point.
 

859
00:53:58 --> 00:54:03
Saddle points are when you have
two field problems, you're

860
00:54:03 --> 00:54:07
talking about saddle points,
and the mixed method in finite

861
00:54:07 --> 00:54:09
elements is exactly that.
 

862
00:54:09 --> 00:54:17
And the tricky part is then
with the mixed method, you're

863
00:54:17 --> 00:54:20
sort of not so perfectly
guaranteed that the

864
00:54:20 --> 00:54:21
matrix is invertible.
 

865
00:54:21 --> 00:54:26
Because we have plot plus
stuff and minus stuff.

866
00:54:26 --> 00:54:28
OK, thank you that's great.
 

867
00:54:28 --> 00:54:33
And I'll see you Monday all
about the exam and review, it's

868
00:54:33 --> 00:54:36
a great chance to think back.