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PROFESSOR STRANG: OK, so this
is a review session with
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open questions on homework.
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Open to questions on
topics in the exam
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that's coming tomorrow.
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This morning I wrote down what
the four questions would be
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about, and I'm glad I did.
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I never should have done
this many times before.
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So you would know exactly
and get down to seeing
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what those problems are.
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And of course the matrices
called K, and A transpose
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C A are going to appear
probably more than once.
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So, open for any questions.
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About any topic whatsoever.
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Please.
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Yes, thank you.
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AUDIENCE: [INAUDIBLE]
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PROFESSOR STRANG: The fourth
question on the exam?
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AUDIENCE: [INAUDIBLE]
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PROFESSOR STRANG: I'm glad
you used that word, fun.
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Yes.
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That's exactly what I mean.
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Section 2.4, and
they are fun, yeah.
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So I drew by hand a little
graph with nodes and edges.
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And you want to be able to
take that first basic step.
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So the first step, which is as
far as we got by last
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Wednesday, the first lecture on
Section 2.4, was just creating
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the matrix A, understanding A
transpose A, and of course
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there's more to understand
about A transpose A.
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Actually, why don't
we take one second.
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Suppose I have a graph with
six nodes, let's say.
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Can you imagine a
graph with six nodes?
40
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And every node connected
to every other node.
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So however many edges
that would be.
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Actually, my grandson just got
that question on his exam.
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He was told there were l
islands with a flight from
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every island to every other
island, and he was asked how
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many flights that makes.
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So I sent him the answer.
47
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But I was very happy
with his reply.
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He said "that's exactly what I
got." So, what do you know.
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It seems to work.
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So anyway.
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Suppose we had, how many
nodes did I say? six?
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OK.
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So we have like a six node, so
n is six, and it's a complete
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graph, this is really just to
start us off talking about
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some of these problems.
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So the matrix A, so I think
it would be 15, where did I
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come up with that number 15?
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And is it right, actually?
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Yes.
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This is one way, would be the
first node has five edges going
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out and then the second node
would have four additional
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edges, and three
and two and one.
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And five, four, three,
two, one add to 15.
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So would be the shape
of A in that case?
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So it has a row for every edge.
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So 15 by six, I think.
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OK, and I could create
A transpose A just to
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have a look at it.
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So it would be, what shape
what A transpose A be?
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Six by six, symmetric,
of course.
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Will it be singular
or non singular?
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Singular.
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Singular, because we haven't
grounded any nodes.
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We've got all these nodes,
all these edges, nothing.
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We haven't taken out that
column; when I reduce it
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to five by five, then
it'll be invertible.
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But six by six, so what will
be the diagonal of this?
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This'll be now six by six, the
the size will be six by six.
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And what will go on the
diagonal is the degrees
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of every node.
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That means how many edges
are coming in, and
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what number is that?
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Five.
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So I'll have five down the
diagonal, and what else, what
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will be off the diagonal?
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Minus, a whole lot of minus
ones, a minus one above
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and below for every edge.
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And since we have a complete
graph, how many minus
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ones have we got?
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All of them.
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All minus ones.
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So all minus ones
and all minus ones.
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That's fine to cross over
if you need to, sure.
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I'm not sure what else to
say about that matrix.
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Well, it's not invertible.
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Now let's take the next step
which, I'm now going probably
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beyond the exam part.
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Just really to get us started,
I ground the six nodes, suppose
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I ground node number six,
that wipes out a row and
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a column, is that right?
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00:05:57 --> 00:06:01
So I'm now left with a
five by five matrix.
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It still has all minus ones
there and there, but now it's
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five by five, now it is what
kind of a matrix, what
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are its properties?
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Square, obviously, symmetric
obviously, and now invertible.
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Positive definitely, OK.
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So it's fives there and now I
would have five minus ones.
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Let's just write them in here.
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Typical row, now in this five
by five matrix would have four
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minus ones and of course more
here, and more here,
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and one there.
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And symmetric.
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All I want to say is that that
matrix, we don't often write
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down the inverses of matrices,
but that one I think we could.
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I think we could actually, and
it's a little bit interesting
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to know, for that special
matrix, everything about it.
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We could find its eigenvalues,
its determinate, its
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pivots, the whole
works for that matrix.
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And that's one page of the
book, maybe at the end of
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Section 2.4, I think, comes
more detail about that matrix.
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So in a way that special guy is
like our special K matrix -1,
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2, -1 for second differences.
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Somehow this is taking,
all nodes are connected.
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Instead of in a line, springs
in a line, points in a
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line, we now have everybody
connected to everybody.
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So this is sort of the special
matrix when everybody is
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connected to everybody and we
could learn all about
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that particular one.
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But then, of course, if some
edges are not in then some
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zeroes will appear off the
diagonal in adjacent
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C matrix part.
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The degrees will drop a little
if we're missing some edges and
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the inverse will be not
some simple expression.
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Anyway, that's to
get us started.
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So that's really where the
last lecture, Friday,
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00:08:14 --> 00:08:16
brought us to this point.
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And I'll take this chance to
add in the block matrix just
138
00:08:21 --> 00:08:24
because I think of it
as quite important.
139
00:08:24 --> 00:08:26
So for this case, C
is the identity.
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So I would have the identity up
in that block, A in that block,
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A transpose in this block.
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00:08:33 --> 00:08:39
That would be my mixed method
matrix, you could say.
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00:08:39 --> 00:08:42
My saddle point matrix.
144
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It starts out very
positive, definite.
145
00:08:45 --> 00:08:48
But it ends up
negative definite.
146
00:08:48 --> 00:08:52
And that's typical of mixed
methods, when both unknowns,
147
00:08:52 --> 00:08:58
the currents as well as the
potentials, are included
148
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in the system.
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So A transpose w, that was
f, I think, and this is b.
150
00:09:04 --> 00:09:09
I just mentioned that again, it
was in Friday's lecture and
151
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it's in the book but I would
just want to say I often refer
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to this as the fundamental
problem of numerical analysis,
153
00:09:17 --> 00:09:20
is how do you solve
that system.
154
00:09:20 --> 00:09:23
And of course elimination
is one way to do it.
155
00:09:23 --> 00:09:28
When I eliminate w, that will
lead me to the equation A
156
00:09:28 --> 00:09:34
transpose Au equals, I think
it'll be, there'll be an A
157
00:09:34 --> 00:09:39
transpose b minus f, I think.
158
00:09:39 --> 00:09:41
C being the identity there.
159
00:09:41 --> 00:09:43
So that's the mixed method,
this is the displacement
160
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method, and this is
the popular one.
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00:09:47 --> 00:09:49
Because it's all at once.
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But people think
about this one, too.
163
00:09:55 --> 00:10:00
So that's like saying what
was in Friday's lecture and
164
00:10:00 --> 00:10:04
will be used going forward.
165
00:10:04 --> 00:10:06
OK, that was just
to get us started.
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00:10:06 --> 00:10:12
Now, please let's
have some questions.
167
00:10:12 --> 00:10:14
We need another question.
168
00:10:14 --> 00:10:15
Who else?
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00:10:15 --> 00:10:16
Yes, thank you.
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00:10:16 --> 00:10:18
AUDIENCE: [INAUDIBLE]
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00:10:18 --> 00:10:19
PROFESSOR STRANG: The first
one in the homework.
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00:10:19 --> 00:10:23
What number was that?
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00:10:23 --> 00:10:27
Section 2.2, number six?
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00:10:27 --> 00:10:30
About the trapezoidal rule?
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00:10:30 --> 00:10:31
Yes.
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00:10:31 --> 00:10:38
OK, now I did speak about that
a little in the last review
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00:10:38 --> 00:10:42
session, but can I just say a
couple words more
178
00:10:42 --> 00:10:43
about it here?
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AUDIENCE: [INAUDIBLE]
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00:10:44 --> 00:10:45
PROFESSOR STRANG:
What's it asking?
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00:10:45 --> 00:10:48
Yes.
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00:10:48 --> 00:10:50
People often ask me that
about my problems.
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00:10:50 --> 00:10:54
I don't know.
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00:10:54 --> 00:10:55
You can't read my mind?
185
00:10:55 --> 00:10:56
You should.
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00:10:56 --> 00:11:04
OK, so the point is that for
special differential equations,
187
00:11:04 --> 00:11:07
so let me just summarize
what we did there.
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00:11:07 --> 00:11:10
So this we did before, but
I didn't do everything.
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So what we did before was point
out that the system du/dt=Au,
190
00:11:18 --> 00:11:27
conserves energy. u squared, u
of time, for all time, u of
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00:11:27 --> 00:11:37
time squared stays constant if
A transpose equals minus A.
192
00:11:37 --> 00:11:43
OK, essentially you take the
derivative, it's got two terms
193
00:11:43 --> 00:11:47
because we've got a product
there, a product rule.
194
00:11:47 --> 00:11:52
The derivative will be, one
term will involve a, the other
195
00:11:52 --> 00:11:56
term will involve A transpose,
if out matrix has this
196
00:11:56 --> 00:11:59
antisymmetric property, those
terms will cancel; the
197
00:11:59 --> 00:12:01
derivative will be zero,
and that'll mean that
198
00:12:01 --> 00:12:03
this is a constant.
199
00:12:03 --> 00:12:06
OK, so that's the
differential equation.
200
00:12:06 --> 00:12:09
Now, the question was about
the difference equation.
201
00:12:09 --> 00:12:15
So we're taking the trapezoidal
rule and we want to show that
202
00:12:15 --> 00:12:22
un squared stays constant
for the trapezoidal rule.
203
00:12:22 --> 00:12:30
And so what that means, in
other words is step by step,
204
00:12:30 --> 00:12:39
U_(n+1), and I could write it
U_(n+1) squared, but the other
205
00:12:39 --> 00:12:43
way to write that and the way
we have to work with it
206
00:12:43 --> 00:12:47
is that is the same.
207
00:12:47 --> 00:12:51
Now, that was just an identity,
that's just the meaning.
208
00:12:51 --> 00:12:56
Now, I want to show
that that's the key.
209
00:12:56 --> 00:12:59
That's what we would
want to prove.
210
00:12:59 --> 00:13:04
That the trapezoidal rule
copies the property of
211
00:13:04 --> 00:13:08
constant energy of the
differential equations.
212
00:13:08 --> 00:13:12
And of course, you know that
in oscillating springs when
213
00:13:12 --> 00:13:17
there's no source, no forces
coming from outside the total
214
00:13:17 --> 00:13:19
energy will stay constant.
215
00:13:19 --> 00:13:22
And you could think of
many other situations.
216
00:13:22 --> 00:13:26
You have a spacecraft, where
you've turned off the engines.
217
00:13:26 --> 00:13:29
It's just going there,
it's possibly rotating.
218
00:13:29 --> 00:13:36
So there you've got angular
velocity included in
219
00:13:36 --> 00:13:37
the total energy.
220
00:13:37 --> 00:13:41
Important fact, if energy stays
constant you want to know it.
221
00:13:41 --> 00:13:44
And you're very happy if
the finite difference
222
00:13:44 --> 00:13:46
method copies it.
223
00:13:46 --> 00:13:54
OK, so then it was just a
question of here we took
224
00:13:54 --> 00:13:56
derivatives to do that one.
225
00:13:56 --> 00:13:59
Here we're going to be playing
with differences, and my
226
00:13:59 --> 00:14:07
suggestion was that the good
way to get it was to take that
227
00:14:07 --> 00:14:16
vector times the trapezoidal
equation and show that this
228
00:14:16 --> 00:14:20
turned out to, the trapezoidal
equation is something equals
229
00:14:20 --> 00:14:25
zero, and you hope, and it
takes a few lines of jiggling
230
00:14:25 --> 00:14:31
around, that when you do that
you'll get the difference.
231
00:14:31 --> 00:14:32
You get exactly this.
232
00:14:32 --> 00:14:39
You get U_(n+1) transpose
U_(n+1) minus U_n
233
00:14:39 --> 00:14:41
transpose U_n.
234
00:14:41 --> 00:14:44
That's the goal.
235
00:14:44 --> 00:14:47
We know that the trapezoidal
equation, maybe I move
236
00:14:47 --> 00:14:51
everything onto one side so I
have something equals zero,
237
00:14:51 --> 00:14:57
then my trick is take that
vector equation, multiply by
238
00:14:57 --> 00:15:03
that, play around with those
terms and you'll get this.
239
00:15:03 --> 00:15:07
So, since that is
zero, this is zero.
240
00:15:07 --> 00:15:11
And that's exactly what
our goal was to prove.
241
00:15:11 --> 00:15:14
So it's just in the jiggling
around and maybe we don't
242
00:15:14 --> 00:15:18
want to take the full time
because I'll post that.
243
00:15:18 --> 00:15:22
Actually, I may post
some of these solutions
244
00:15:22 --> 00:15:24
even before the quiz.
245
00:15:24 --> 00:15:27
And therefore before the
homework is due, just because
246
00:15:27 --> 00:15:32
this particular homework, as I
say, is not, the graders are
247
00:15:32 --> 00:15:39
just going to be so busy
with all the quizzes.
248
00:15:39 --> 00:15:40
This is for learning.
249
00:15:40 --> 00:15:43
Now, here's the one thing
you want to learn out of
250
00:15:43 --> 00:15:46
this messy computation.
251
00:15:46 --> 00:15:55
And also have a term, you'll
also find a term U_n, when you
252
00:15:55 --> 00:15:59
just do this mechanically,
you'll find a U_(n+1) transpose
253
00:15:59 --> 00:16:07
U_n, and you'll find a U_n
transpose U_(n+1), and they'll
254
00:16:07 --> 00:16:10
come in with opposite signs.
255
00:16:10 --> 00:16:13
That'll be when you've plugged
in the fact that A transpose
256
00:16:13 --> 00:16:19
equals minus A, and all I
wanted to do is ask you, what
257
00:16:19 --> 00:16:20
does that term amount to?
258
00:16:20 --> 00:16:23
Because that term will show up.
259
00:16:23 --> 00:16:24
One way or another.
260
00:16:24 --> 00:16:26
And what does it equal?
261
00:16:26 --> 00:16:27
Zero.
262
00:16:27 --> 00:16:28
Everybody should know that.
263
00:16:28 --> 00:16:32
That's the one thing, if you
have to add to just
264
00:16:32 --> 00:16:37
mechanically computing is the
fact that the dot product of
265
00:16:37 --> 00:16:42
that vector with that, v
transpose w is the same
266
00:16:42 --> 00:16:46
as w transpose v.
267
00:16:46 --> 00:16:49
So, it's good to just call
attention to the easy
268
00:16:49 --> 00:16:51
things that are like that.
269
00:16:51 --> 00:16:53
Why is that?
270
00:16:53 --> 00:16:59
That's because both sides, this
is equal to what? v_1*w_1,
271
00:16:59 --> 00:17:03
v_2*w_2, v_3*w_3,
component by component.
272
00:17:03 --> 00:17:04
And this is w_1*v_1.
273
00:17:05 --> 00:17:08
But we're just talking
numbers at that point.
274
00:17:08 --> 00:17:10
So the numbers of v_1 times
w_1 are certainly the
275
00:17:10 --> 00:17:12
same as w_1 times v_1.
276
00:17:12 --> 00:17:16
Every component by component,
they're exactly the same and
277
00:17:16 --> 00:17:19
of course then the dot
products are the same.
278
00:17:19 --> 00:17:27
So that's the fact which for
this v and that w, make the
279
00:17:27 --> 00:17:31
term go away, that's
still sitting there.
280
00:17:31 --> 00:17:35
Other terms go away
because of this property.
281
00:17:35 --> 00:17:40
Having written that and
recognizing that we have
282
00:17:40 --> 00:17:46
Fourier stuff coming up in the
last third of the course, where
283
00:17:46 --> 00:17:48
we have complex numbers.
284
00:17:48 --> 00:17:57
I have to say that if when
I have complex vectors,
285
00:17:57 --> 00:18:01
do you know about those?
286
00:18:01 --> 00:18:06
The dot product, or the length
squared, if this was a vector
287
00:18:06 --> 00:18:10
of complex, with possibly
complex numbers, I wouldn't
288
00:18:10 --> 00:18:14
take the length squared just
by adding up these squares.
289
00:18:14 --> 00:18:19
Suppose, my, yes, I'm really
anticipating weeks ahead, but
290
00:18:19 --> 00:18:23
suppose my vector was .
291
00:18:23 --> 00:18:27
What's the length of that
particular vector, v?
292
00:18:27 --> 00:18:31
Well, if I do v transpose
v, what do I get?
293
00:18:31 --> 00:18:36
For v equals , what does
v transpose v turn out to be?
294
00:18:36 --> 00:18:37
Zero.
295
00:18:37 --> 00:18:40
One squared plus i
squared is zero.
296
00:18:40 --> 00:18:41
No good.
297
00:18:41 --> 00:18:45
So obviously some rule has
to change a little bit to
298
00:18:45 --> 00:18:46
get the correct number.
299
00:18:46 --> 00:18:51
The correct length squared,
I would rather expect two.
300
00:18:51 --> 00:18:54
The size of that squared plus
the size of that squared.
301
00:18:54 --> 00:18:56
So I don't want to square
i, I want to square
302
00:18:56 --> 00:18:58
its absolute value.
303
00:18:58 --> 00:19:01
And the way to do that
is conjugate one
304
00:19:01 --> 00:19:03
of the two things.
305
00:19:03 --> 00:19:08
Now I'm taking , and on
the other side I have
306
00:19:08 --> 00:19:11
and that gives me the
two that I want.
307
00:19:11 --> 00:19:18
So what I'm doing, when I've
got complex vectors then I
308
00:19:18 --> 00:19:23
would really do that, and now
that is not the same as that.
309
00:19:23 --> 00:19:24
Right. yeah.
310
00:19:24 --> 00:19:31
If in one case if I'm doing the
conjugate of v and the other
311
00:19:31 --> 00:19:33
case it's the conjugate of
w, then I've got a
312
00:19:33 --> 00:19:35
complex conjugate.
313
00:19:35 --> 00:19:41
OK, that's a throwaway comment
that just is relevant because
314
00:19:41 --> 00:19:47
it keeps us focused for a
moment on the real case,
315
00:19:47 --> 00:19:52
where we do have equals.
316
00:19:52 --> 00:19:55
Now, I don't know if that
was sufficient answer?
317
00:19:55 --> 00:19:59
It wasn't a complete answer
because I didn't do the
318
00:19:59 --> 00:20:04
manipulations, but the
solutions posted should
319
00:20:04 --> 00:20:06
show you those.
320
00:20:06 --> 00:20:08
And, of course, you
can organize them a
321
00:20:08 --> 00:20:09
little different.
322
00:20:09 --> 00:20:11
OK, good for that one.
323
00:20:11 --> 00:20:12
Yes, please.
324
00:20:12 --> 00:20:16
AUDIENCE: [INAUDIBLE]
325
00:20:16 --> 00:20:18
PROFESSOR STRANG: The
other two terms here?
326
00:20:18 --> 00:20:22
AUDIENCE: [INAUDIBLE]
327
00:20:22 --> 00:20:25
PROFESSOR STRANG: Yes.
328
00:20:25 --> 00:20:28
You couldn't cancel them.
329
00:20:28 --> 00:20:34
Well, I recommend just, that's
how the best mathematics
330
00:20:34 --> 00:20:35
is done, right?
331
00:20:35 --> 00:20:39
You want zero, you
just x it out.
332
00:20:39 --> 00:20:40
Anyway.
333
00:20:40 --> 00:20:44
Let me leave the posted
solutions to be a hint
334
00:20:44 --> 00:20:46
and come back to it.
335
00:20:46 --> 00:20:47
Yeah.
336
00:20:47 --> 00:20:52
AUDIENCE: [INAUDIBLE]
337
00:20:52 --> 00:20:54
PROFESSOR STRANG: The
next problem was?
338
00:20:54 --> 00:20:56
AUDIENCE: [INAUDIBLE]
339
00:20:56 --> 00:20:58
PROFESSOR STRANG: Oh, yes.
340
00:20:58 --> 00:21:03
OK, that one I spoke a little
bit about, but now let me read
341
00:21:03 --> 00:21:08
from the problem
set that I got.
342
00:21:08 --> 00:21:12
I noticed that was quite brief.
343
00:21:12 --> 00:21:16
Oh, to find that actual angle?
344
00:21:16 --> 00:21:20
Somehow that's a little
interesting, isn't it?
345
00:21:20 --> 00:21:24
AUDIENCE: [INAUDIBLE]
346
00:21:24 --> 00:21:28
PROFESSOR STRANG: To tell the
truth, I meant numerically.
347
00:21:28 --> 00:21:33
I meant, what's the
point of that question.
348
00:21:33 --> 00:21:38
The point is we're trying to
solve, this isn't a big deal.
349
00:21:38 --> 00:21:44
But it was just if we're using
this trapezoidal method, the
350
00:21:44 --> 00:21:50
beauty of that exactly what our
thing proves is, here the
351
00:21:50 --> 00:21:54
constant energy surface
is the circle.
352
00:21:54 --> 00:21:57
The point of the trapezoidal
method for this simple equation
353
00:21:57 --> 00:22:07
u''+u=0, which amounted to the
equation uv' equals something
354
00:22:07 --> 00:22:17
like, our a matrix
was antisymmetric.
355
00:22:17 --> 00:22:20
So it fit perfectly in
that problem, and if we
356
00:22:20 --> 00:22:23
started on the circle
we stay on the circle.
357
00:22:23 --> 00:22:28
And if I take 32 steps I come
back pretty closely to here,
358
00:22:28 --> 00:22:33
and I just thought it might be
fun to figure out numerically,
359
00:22:33 --> 00:22:38
with MATLAB or a calculator or
something, we take an angle,
360
00:22:38 --> 00:22:43
theta, is that what the problem
asks, and then come around here
361
00:22:43 --> 00:22:49
to 32 theta, and 32 theta
will not be exactly 2pi.
362
00:22:49 --> 00:22:51
But darned close.
363
00:22:51 --> 00:22:53
Because you could see in
the figure in the book
364
00:22:53 --> 00:22:56
it wasn't too far off.
365
00:22:56 --> 00:22:59
So the question was,
what is that theta?
366
00:22:59 --> 00:23:05
So I think that the formula
turned out to be that each step
367
00:23:05 --> 00:23:12
multiplied by this one plus i
delta t, or h on two divided by
368
00:23:12 --> 00:23:15
one minus i delta t over two.
369
00:23:15 --> 00:23:24
And when you plug in delta t to
be 2pi over 32, so that's the,
370
00:23:24 --> 00:23:28
what did I say, that's the
tangent of theta or something?
371
00:23:28 --> 00:23:32
Sorry, I've forgotten the
way the problem was put.
372
00:23:32 --> 00:23:38
Oh, it's e to the
i theta, yeah.
373
00:23:38 --> 00:23:41
What's the main point about
that complex number?
374
00:23:41 --> 00:23:44
When you look at that complex
number what's the most
375
00:23:44 --> 00:23:47
essential thing you see?
376
00:23:47 --> 00:23:49
That it, yeah, tell me again.
377
00:23:49 --> 00:23:50
AUDIENCE: [INAUDIBLE]
378
00:23:50 --> 00:23:52
PROFESSOR STRANG:
Magnitude one, great.
379
00:23:52 --> 00:23:55
It's a number divided by its
complex conjugate, so it's
380
00:23:55 --> 00:23:57
a number of magnitude one.
381
00:23:57 --> 00:24:01
And now tell me, if you see a
complex number of magnitude
382
00:24:01 --> 00:24:03
one, what jumps to mind?
383
00:24:03 --> 00:24:07
What form do you naturally
put it in? e^(i*theta).
384
00:24:07 --> 00:24:10
385
00:24:10 --> 00:24:14
Every complex number of
absolute value one is just
386
00:24:14 --> 00:24:16
beautifully written in
the form e^(i*theta).
387
00:24:16 --> 00:24:19
388
00:24:19 --> 00:24:23
That complex number is
some point on the unit
389
00:24:23 --> 00:24:24
circle, so there it is.
390
00:24:24 --> 00:24:26
Right there, there
it is, e^(i*theta).
391
00:24:28 --> 00:24:32
With that, theta is negative
there, because we're
392
00:24:32 --> 00:24:33
going the wrong way.
393
00:24:33 --> 00:24:35
No big deal.
394
00:24:35 --> 00:24:38
Maybe here theta's positive.
395
00:24:38 --> 00:24:45
I've forgotten, so I won't try
to go either the clockwise or
396
00:24:45 --> 00:24:47
the counterclockwise
way around.
397
00:24:47 --> 00:24:51
So, if I wanted to figure out
what theta was and plugged
398
00:24:51 --> 00:24:54
in these things, let's see.
399
00:24:54 --> 00:25:03
So that's pi over 32, delta t
over two would be pi over 32,
400
00:25:03 --> 00:25:11
and this guy would be its
conjugate. pi over 32, and then
401
00:25:11 --> 00:25:16
in this solution that'll be
plugged on the homework this
402
00:25:16 --> 00:25:28
will be, I think maybe, the
theta comes out to be, it's
403
00:25:28 --> 00:25:34
kind of cool actually, twice
the arc tangent of pi
404
00:25:34 --> 00:25:37
over 32 or something.
405
00:25:37 --> 00:25:38
I didn't know that.
406
00:25:38 --> 00:25:44
But that'll be in the
solutions for you to check.
407
00:25:44 --> 00:25:50
So now, why do I like
e^(i*theta) so much?
408
00:25:50 --> 00:25:54
Because now I could tell you
what this point is, after
409
00:25:54 --> 00:25:56
you've done it 32 times.
410
00:25:56 --> 00:25:59
What's angle have you reached?
411
00:25:59 --> 00:26:04
This is the crunch line of
using complex numbers,
412
00:26:04 --> 00:26:08
e^(i*theta), is that
they're absolutely great
413
00:26:08 --> 00:26:09
for taking powers.
414
00:26:09 --> 00:26:15
If I take the 32nd power of x
plus iy, I'm lost, right. x
415
00:26:15 --> 00:26:20
plus iy to the 32nd power
starts out x^(32), ends
416
00:26:20 --> 00:26:24
up i^(32), y^(32), with
horrible stuff in between.
417
00:26:24 --> 00:26:28
But what is the 32nd
power of e^(i*theta)?
418
00:26:30 --> 00:26:30
e^(i*32theta).
419
00:26:30 --> 00:26:33
420
00:26:33 --> 00:26:37
Just that angle 32 times
exactly as we've drawn it.
421
00:26:37 --> 00:26:39
So that's the point
e^(i*32theta).
422
00:26:39 --> 00:26:42
423
00:26:42 --> 00:26:49
OK, and therefore if we now
know what theta is, so yeah.
424
00:26:49 --> 00:26:55
So it must be pretty near
2pi, but not exactly.
425
00:26:55 --> 00:26:59
I guess that's about right.
426
00:26:59 --> 00:27:04
In fact, having got this far,
the tangent of a very small
427
00:27:04 --> 00:27:09
angle is approximately what?
428
00:27:09 --> 00:27:11
It's approximately
the angle, right?
429
00:27:11 --> 00:27:14
The sine of a very small angle
is approximately the angle.
430
00:27:14 --> 00:27:16
The cosine is
approximately one.
431
00:27:16 --> 00:27:19
The tangent is
approximately the angle.
432
00:27:19 --> 00:27:26
So this, 32 theta, is 32 times
two times approximately
433
00:27:26 --> 00:27:28
the angle.
434
00:27:28 --> 00:27:33
And what answer do you get?
435
00:27:33 --> 00:27:35
2pi.
436
00:27:35 --> 00:27:38
Which makes sense.
437
00:27:38 --> 00:27:47
So you could say what the
trapezoidal method has done is
438
00:27:47 --> 00:27:52
to replace the exact angle by
the inverse tangent
439
00:27:52 --> 00:27:53
approximately.
440
00:27:53 --> 00:27:56
That's sort of nice.
441
00:27:56 --> 00:27:59
In this example you can get
as far as that and you
442
00:27:59 --> 00:28:03
could actually find
out how near that is.
443
00:28:03 --> 00:28:06
And, by the way, how near
what I expected it to be.
444
00:28:06 --> 00:28:11
I would expect it, so what
do we know about the
445
00:28:11 --> 00:28:16
trapezoidal method
without having proved it?
446
00:28:16 --> 00:28:19
It's second order
accurate, right?
447
00:28:19 --> 00:28:23
If it was first order accurate,
I would expect it to miss by
448
00:28:23 --> 00:28:26
something of the size of theta.
449
00:28:26 --> 00:28:28
Maybe a fraction of theta.
450
00:28:28 --> 00:28:30
But being second order
accurate, I'm expecting
451
00:28:30 --> 00:28:34
it to miss by something
of size theta squared.
452
00:28:34 --> 00:28:40
So it would be pretty
near zero, right.
453
00:28:40 --> 00:28:46
And actually, another way I
know it, around, and so the
454
00:28:46 --> 00:28:50
error would be something like,
it would have a 32 squared
455
00:28:50 --> 00:28:51
in the denominator.
456
00:28:51 --> 00:28:59
And I've just thought of
another way to see that.
457
00:28:59 --> 00:29:02
We just said that the first
term in the arc tangent of a
458
00:29:02 --> 00:29:07
small angle, theta, of a small
angle, alpha, whatever that is,
459
00:29:07 --> 00:29:11
pi over 32, the first term
in the arc tangent is?
460
00:29:11 --> 00:29:12
The angle.
461
00:29:12 --> 00:29:14
That's what we just said.
462
00:29:14 --> 00:29:16
Then, do you know what
would come next?
463
00:29:16 --> 00:29:20
Now we're looking at the error.
464
00:29:20 --> 00:29:27
So that of a very small angle
will start theta, and I want to
465
00:29:27 --> 00:29:32
ask you about how many theta
squareds and theta cubes.
466
00:29:32 --> 00:29:38
You're seeing what you
can do with paper and
467
00:29:38 --> 00:29:41
pencils type stuff.
468
00:29:41 --> 00:29:47
Here's my main question, how
many theta squareds in there?
469
00:29:47 --> 00:29:48
You want to make a guess?
470
00:29:48 --> 00:29:51
A mathematician's
favorite number, zero.
471
00:29:51 --> 00:29:55
Right, there will be no
theta squared terms in.
472
00:29:55 --> 00:29:59
That's an odd function, so I'm
expecting only odd powers and
473
00:29:59 --> 00:30:02
therefore I won't be surprised
to see theta cubed
474
00:30:02 --> 00:30:04
come up first.
475
00:30:04 --> 00:30:09
And then when I multiply by the
32, I get the theta squared
476
00:30:09 --> 00:30:14
that I guessed we would have.
477
00:30:14 --> 00:30:22
OK, once again I'll stop there
because that's very narrow path
478
00:30:22 --> 00:30:26
to be following but
it shows you how.
479
00:30:26 --> 00:30:30
You know, there's a lot of room
still for what you can do
480
00:30:30 --> 00:30:35
with paper and pencil to
understand a model problem.
481
00:30:35 --> 00:30:40
And then the computer would
tell us about a serious problem
482
00:30:40 --> 00:30:44
of following the solar
system for a million years.
483
00:30:44 --> 00:30:48
OK, another totally different
question, if I can.
484
00:30:48 --> 00:30:49
Yes, thank you.
485
00:30:49 --> 00:30:50
AUDIENCE: [INAUDIBLE]
486
00:30:50 --> 00:30:51
PROFESSOR STRANG: Yeah, sure.
487
00:30:51 --> 00:30:53
AUDIENCE: [INAUDIBLE]
488
00:30:53 --> 00:30:56
PROFESSOR STRANG: 2.4.1, right.
489
00:30:56 --> 00:30:57
A mistake in the book.
490
00:30:57 --> 00:30:59
AUDIENCE: [INAUDIBLE]
491
00:30:59 --> 00:31:01
PROFESSOR STRANG:
Or in the, yeah.
492
00:31:01 --> 00:31:04
It's quite possible.
493
00:31:04 --> 00:31:12
OK, there's a printed
error in the graph.
494
00:31:12 --> 00:31:13
Yeah.
495
00:31:13 --> 00:31:18
So in numbering the edges,
well let's blame it on
496
00:31:18 --> 00:31:19
the printer, right?
497
00:31:19 --> 00:31:20
Not the author.
498
00:31:20 --> 00:31:26
OK, so the diagonal edge, that
five probably was intended
499
00:31:26 --> 00:31:28
to be a three, yeah.
500
00:31:28 --> 00:31:29
Thank you.
501
00:31:29 --> 00:31:35
So we'll catch that in
the next printing.
502
00:31:35 --> 00:31:42
And you recognize that always
numbering the edges and nodes
503
00:31:42 --> 00:31:45
is a pretty arbitrary thing,
it's just if we number
504
00:31:45 --> 00:31:50
differently that just reorders
the rows of the matrix.
505
00:31:50 --> 00:31:52
If we number the edge
differently, it'll reorder the
506
00:31:52 --> 00:31:57
rows and it'll reorder rows and
columns of A transpose A.
507
00:31:57 --> 00:32:00
So it won't make a serious
difference in the matrix.
508
00:32:00 --> 00:32:00
Yeah.
509
00:32:00 --> 00:32:03
AUDIENCE: [INAUDIBLE]
510
00:32:03 --> 00:32:05
PROFESSOR STRANG: Do you want
to go back to this guy?
511
00:32:05 --> 00:32:06
OK.
512
00:32:06 --> 00:32:11
AUDIENCE: So if you have an
anti-symmetric matrix, does it
513
00:32:11 --> 00:32:12
follow that the eigenvectors
used are perpendicular?
514
00:32:12 --> 00:32:16
PROFESSOR STRANG: This
is a good question.
515
00:32:16 --> 00:32:20
This guy, way up here,
with this property,
516
00:32:20 --> 00:32:22
AUDIENCE: The eigenvectors
are perpendicular?
517
00:32:22 --> 00:32:24
PROFESSOR STRANG: The
eigenvectors are perpendicular.
518
00:32:24 --> 00:32:26
Yes, yeah.
519
00:32:26 --> 00:32:32
So we have, there's this, like,
the nobility among matrices are
520
00:32:32 --> 00:32:35
the ones with perpendicular
eigenvectors.
521
00:32:35 --> 00:32:41
So that includes symmetric
matrices, this is a good
522
00:32:41 --> 00:32:43
and straightforward point.
523
00:32:43 --> 00:32:46
So these are the good matrices.
524
00:32:46 --> 00:32:48
Symmetric matrices.
525
00:32:48 --> 00:32:51
A transpose equals A.
526
00:32:51 --> 00:32:56
Their eigenvalues lie
on the real line.
527
00:32:56 --> 00:33:04
And these are all
perpendicular eigenvectors.
528
00:33:04 --> 00:33:08
What about antisymmetric?
529
00:33:08 --> 00:33:12
OK, that means A
transpose is minus A.
530
00:33:12 --> 00:33:16
They also fall in this noble
family of matrices, and where
531
00:33:16 --> 00:33:19
are their eigenvalues?
532
00:33:19 --> 00:33:22
Pure imaginary, right up here.
533
00:33:22 --> 00:33:26
Now do you want to know, who
else is in this family?
534
00:33:26 --> 00:33:29
What's the other, this is the
complex plane; there's one more
535
00:33:29 --> 00:33:32
piece of the complex plane that
you know I'm going to put.
536
00:33:32 --> 00:33:35
Which is?
537
00:33:35 --> 00:33:39
What else to make that complex
plane look familiar, it's going
538
00:33:39 --> 00:33:43
to have the unit circle.
539
00:33:43 --> 00:33:46
Every complex plane has got
to have the unit circle.
540
00:33:46 --> 00:33:51
OK, so these guys went with
the, and now what do you
541
00:33:51 --> 00:33:55
think goes with the, this
will be the matrices.
542
00:33:55 --> 00:33:57
Can I call them Q instead,
because I called
543
00:33:57 --> 00:33:59
them Q this morning.
544
00:33:59 --> 00:34:02
Q transpose is Q inverse.
545
00:34:02 --> 00:34:05
Q transpose Q, and they're
the orthogonal matrices.
546
00:34:05 --> 00:34:10
So those matrices again,
beautiful matrices
547
00:34:10 --> 00:34:11
in the best class.
548
00:34:11 --> 00:34:15
And their eigenvalues
are on the unit circle.
549
00:34:15 --> 00:34:17
And that would be z.
550
00:34:17 --> 00:34:20
Why don't I just show you why?
551
00:34:20 --> 00:34:23
Because orthogonal matrices,
there are not so many that
552
00:34:23 --> 00:34:25
are really worth knowing.
553
00:34:25 --> 00:34:31
So, let me take Qx=lambda*x,
and what is it that
554
00:34:31 --> 00:34:33
I want to prove?
555
00:34:33 --> 00:34:37
I want to prove that
the eigenvalues have
556
00:34:37 --> 00:34:39
absolute value one.
557
00:34:39 --> 00:34:40
That's the unit circle.
558
00:34:40 --> 00:34:42
So how do I show that
the eigenvalues have
559
00:34:42 --> 00:34:45
absolute value of one?
560
00:34:45 --> 00:34:49
Let me take the dot product
with Qx transpose.
561
00:34:49 --> 00:34:56
So both sides, I'll do Qx
transpose Qx and I'll do
562
00:34:56 --> 00:35:01
lambda*x transpose
lambda*x, right?
563
00:35:01 --> 00:35:04
Only these are complex.
564
00:35:04 --> 00:35:08
I've got to take complex stuff.
565
00:35:08 --> 00:35:10
OK.
566
00:35:10 --> 00:35:17
I just took the length squared
of both sides, and kept in mind
567
00:35:17 --> 00:35:20
the possibility that this x and
lambda could be, and probably
568
00:35:20 --> 00:35:22
will be, complex numbers.
569
00:35:22 --> 00:35:25
Now, what do I
have on the left?
570
00:35:25 --> 00:35:28
Do you see it happening?
571
00:35:28 --> 00:35:31
I get an x bar transpose,
what do I get?
572
00:35:31 --> 00:35:37
Q transpose Qx on
the left side.
573
00:35:37 --> 00:35:40
That's the combination
I'm looking for.
574
00:35:40 --> 00:35:42
For an orthogonal matrix.
575
00:35:42 --> 00:35:45
Let's imagine the matrix
itself is real, otherwise
576
00:35:45 --> 00:35:48
I would just conjugate it.
577
00:35:48 --> 00:35:53
What nice about that left side?
578
00:35:53 --> 00:35:57
What fact am I going
to use about Q?
579
00:35:57 --> 00:36:00
Q transpose Q is the identity.
580
00:36:00 --> 00:36:04
So this thing is nothing
but x bar transpose x.
581
00:36:04 --> 00:36:05
That's the length of x squared.
582
00:36:05 --> 00:36:07
What have I got on
the right side?
583
00:36:07 --> 00:36:11
I've got the length of x
squared times the number.
584
00:36:11 --> 00:36:16
Lambda bar times
lambda squared.
585
00:36:16 --> 00:36:17
It's there, now.
586
00:36:17 --> 00:36:20
On the left side I have
the length of x squared.
587
00:36:20 --> 00:36:23
On the right side I have the
length of x squared times that
588
00:36:23 --> 00:36:25
number, mod lambda squared.
589
00:36:25 --> 00:36:28
Therefore, that number has to
be one and the eigenvalues
590
00:36:28 --> 00:36:30
are on the unit circle.
591
00:36:30 --> 00:36:36
So, I've given you the three
big important classes of
592
00:36:36 --> 00:36:41
matrices with perpendicular
eigenvectors.
593
00:36:41 --> 00:36:46
I think anybody would
wonder, OK, what about
594
00:36:46 --> 00:36:48
other eigenvalues.
595
00:36:48 --> 00:36:52
What's the condition for
perpendicular eigenvectors
596
00:36:52 --> 00:36:53
that includes this.
597
00:36:53 --> 00:36:55
And includes this.
598
00:36:55 --> 00:36:59
And includes this, and
also allows eigenvalues
599
00:36:59 --> 00:37:01
all over the place.
600
00:37:01 --> 00:37:04
Would you like to
know that condition?
601
00:37:04 --> 00:37:06
What the heck.
602
00:37:06 --> 00:37:12
That condition, that includes
all these is this, that A
603
00:37:12 --> 00:37:17
transpose times A equals
A times A transpose.
604
00:37:17 --> 00:37:22
That's the test for
perpendicular eigenvectors.
605
00:37:22 --> 00:37:24
A transpose commutes with A.
606
00:37:24 --> 00:37:26
So this passes, of course.
607
00:37:26 --> 00:37:28
This passes, of course.
608
00:37:28 --> 00:37:32
This passes because both sides
are the identity, and then
609
00:37:32 --> 00:37:34
some more matrices pass also.
610
00:37:34 --> 00:37:36
OK.
611
00:37:36 --> 00:37:37
Is that alright?
612
00:37:37 --> 00:37:41
You asked for some linear
algebra and you got it.
613
00:37:41 --> 00:37:42
Now I'm ready, yes, thanks.
614
00:37:42 --> 00:37:45
AUDIENCE: [INAUDIBLE]
615
00:37:45 --> 00:37:50
PROFESSOR STRANG: 2.4.19.
616
00:37:50 --> 00:37:52
Oh, let me look.
617
00:37:52 --> 00:37:54
2.4.19.
618
00:37:54 --> 00:37:59
Ah.
619
00:37:59 --> 00:38:06
OK, yes, sorry and I should
have done better with that.
620
00:38:06 --> 00:38:23
So one graphs that are
important are grids like this.
621
00:38:23 --> 00:38:28
And we'll see them two, three,
four, one, two, three, four.
622
00:38:28 --> 00:38:32
That would be where,
these are the nodes.
623
00:38:32 --> 00:38:38
So this is a grid.
624
00:38:38 --> 00:38:41
I meant to draw them all in,
but I won't. n squared.
625
00:38:41 --> 00:38:47
So n is six, and
I'd have 36 nodes.
626
00:38:47 --> 00:38:50
And you can see the
edges in there.
627
00:38:50 --> 00:38:53
So that's the graph
I have in mind.
628
00:38:53 --> 00:39:00
And the reason that problem is
there is that last year, I
629
00:39:00 --> 00:39:05
think it was last year or the
year before, we spent some time
630
00:39:05 --> 00:39:09
with figuring out resistances
and currents and so on
631
00:39:09 --> 00:39:11
for these problems.
632
00:39:11 --> 00:39:16
And we needed some fast way to
generate A, because this matrix
633
00:39:16 --> 00:39:19
A is now, what size
is the matrix A?
634
00:39:19 --> 00:39:23
It's got, I don't know how
many edges does it have?
635
00:39:23 --> 00:39:27
One, two, three, four,
five, maybe 30 edges going
636
00:39:27 --> 00:39:29
across and 30 coming down.
637
00:39:29 --> 00:39:38
It'll be 60 by how many
columns in this matrix?
638
00:39:38 --> 00:39:41
You know the answer now and
it's worth knowing, for
639
00:39:41 --> 00:39:42
the quiz of course.
640
00:39:42 --> 00:39:44
36.
641
00:39:44 --> 00:39:48
OK.
642
00:39:48 --> 00:39:55
Anyway, that class rebelled
about creating these matrices
643
00:39:55 --> 00:40:03
and working with the matrices,
with 2,160 non-zeroes.
644
00:40:03 --> 00:40:04
People were dropping
the course.
645
00:40:04 --> 00:40:10
So we need a command that would
create A pretty quickly.
646
00:40:10 --> 00:40:14
And so that's what the
book, and so this was
647
00:40:14 --> 00:40:17
like the 18.085 command.
648
00:40:17 --> 00:40:23
After we stumbled around for a
while, we discovered that a
649
00:40:23 --> 00:40:29
MATLAB command called kron
was a quick way to
650
00:40:29 --> 00:40:31
create the matrix.
651
00:40:31 --> 00:40:38
We'll see that when we
get to that point.
652
00:40:38 --> 00:40:40
This is an important graph.
653
00:40:40 --> 00:40:44
And it's closely
connected to Laplace's.
654
00:40:44 --> 00:40:45
You remember Laplace's?
655
00:40:45 --> 00:40:47
I'll just tell you.
656
00:40:47 --> 00:40:52
Laplace's equation is this, you
have a second x derivative, we
657
00:40:52 --> 00:40:54
know how to deal with those.
658
00:40:54 --> 00:40:59
But it also has a
second y derivative.
659
00:40:59 --> 00:41:02
So I'm really looking ahead at
the most important equation of
660
00:41:02 --> 00:41:05
Chapter 3, Laplace's equation.
661
00:41:05 --> 00:41:11
And suppose I use finite
differences. i want a
662
00:41:11 --> 00:41:16
matrix K2D that deals
with this 2D problem.
663
00:41:16 --> 00:41:19
And let me just say
what it would be.
664
00:41:19 --> 00:41:22
At a typical point this x
derivative is giving me a minus
665
00:41:22 --> 00:41:25
one, a two and a minus one.
666
00:41:25 --> 00:41:28
And the y derivative is giving
me a minus one that moves this
667
00:41:28 --> 00:41:34
guy up to four and minus one.
668
00:41:34 --> 00:41:40
So instead of -1, 2, -1 along a
typical row, we'll now have a
669
00:41:40 --> 00:41:45
four on the diagonal and four
minus one in a certain pattern.
670
00:41:45 --> 00:41:46
Anyway.
671
00:41:46 --> 00:41:50
You'll see that, it's
interesting when we get to it.
672
00:41:50 --> 00:41:53
That would show up
in A transpose A.
673
00:41:53 --> 00:41:59
So what I've said here is what
happens with A transpose A.
674
00:41:59 --> 00:42:04
I guess I'm hoping that you
begin to know these matrices,
675
00:42:04 --> 00:42:08
first seeing them occasionally
in homeworks and
676
00:42:08 --> 00:42:10
then in the lecture.
677
00:42:10 --> 00:42:11
Good.
678
00:42:11 --> 00:42:15
But that's looking ahead.
679
00:42:15 --> 00:42:20
I needed some questions that
are like, close to really on
680
00:42:20 --> 00:42:24
what we're doing or what
the quiz would do.
681
00:42:24 --> 00:42:25
Any - thank you.
682
00:42:25 --> 00:42:30
AUDIENCE: [INAUDIBLE]
683
00:42:30 --> 00:42:33
PROFESSOR STRANG: Oh yes.
684
00:42:33 --> 00:42:35
A little bit.
685
00:42:35 --> 00:42:39
OK, yeah.
686
00:42:39 --> 00:42:44
So I wrote down this equation
and what I'm writing right
687
00:42:44 --> 00:42:47
there is the new part.
688
00:42:47 --> 00:42:52
Sort of new, and I guess equals
some right hand side f(x).
689
00:42:52 --> 00:43:00
And the discrete version will
be an A transpose C A equal a
690
00:43:00 --> 00:43:05
vector f, maybe there will
be a delta x squared here.
691
00:43:05 --> 00:43:07
OK.
692
00:43:07 --> 00:43:12
I guess maybe, I don't want to
go far but I want you to see
693
00:43:12 --> 00:43:16
that if we have a coefficient C
in here it should
694
00:43:16 --> 00:43:20
show up there.
695
00:43:20 --> 00:43:26
You could actually, it might be
reasonable to look at 3.1 just
696
00:43:26 --> 00:43:31
to look slightly ahead to see
the parallels but you would get
697
00:43:31 --> 00:43:37
them right without
a lecture on it.
698
00:43:37 --> 00:43:42
Your coefficient shows up in
the differential equation, and
699
00:43:42 --> 00:43:49
it shows up on the diagonal of
C in the difference equation.
700
00:43:49 --> 00:43:57
I won't give a whole lecture
on that, just to say that
701
00:43:57 --> 00:44:01
correspondence is exactly
the one we know.
702
00:44:01 --> 00:44:05
A is a difference matrix.
703
00:44:05 --> 00:44:07
Like the derivative.
704
00:44:07 --> 00:44:11
C will be a diagonal matrix,
A transpose will be whatever
705
00:44:11 --> 00:44:14
that comes out to be.
706
00:44:14 --> 00:44:18
And so you've seen A
transpose A, but think again
707
00:44:18 --> 00:44:19
about that difference.
708
00:44:19 --> 00:44:22
And ask yourselves this.
709
00:44:22 --> 00:44:27
I suggest, take C to be
one, get C out of there.
710
00:44:27 --> 00:44:33
And just think, again, what is
the difference matrix A with a
711
00:44:33 --> 00:44:38
boundary either fixed-fixed or
fixed-free, those will
712
00:44:38 --> 00:44:40
be two different A's.
713
00:44:40 --> 00:44:53
What are the A's for
fixed-fixed and for fixed-free?
714
00:44:53 --> 00:44:57
This is what we were doing
at the very beginning
715
00:44:57 --> 00:44:59
of the course.
716
00:44:59 --> 00:45:04
So A is a first difference
matrix, and A transpose A will
717
00:45:04 --> 00:45:05
be the second difference.
718
00:45:05 --> 00:45:11
So the A transpose A, of
course, I was doing A transpose
719
00:45:11 --> 00:45:16
A, then the answer here would
be the matrix K and the answer
720
00:45:16 --> 00:45:20
here would be the matrix T.
721
00:45:20 --> 00:45:26
Or, depending which end is
free, but we'd have one change.
722
00:45:26 --> 00:45:30
That's A transpose A,
but now think about the
723
00:45:30 --> 00:45:32
a that it came from.
724
00:45:32 --> 00:45:39
So A will be, A is the matrix
that takes differences of the
725
00:45:39 --> 00:45:45
u's, and then A transpose a
takes second differences.
726
00:45:45 --> 00:45:49
Of all the questions asked,
this is the one most relevant
727
00:45:49 --> 00:45:57
for the exam and for
what we've done so far.
728
00:45:57 --> 00:46:01
I've gone off onto topics
that we look ahead to,
729
00:46:01 --> 00:46:03
but this is where we are.
730
00:46:03 --> 00:46:07
So that matrix a is a first
difference matrix, and then you
731
00:46:07 --> 00:46:11
put in the boundary conditions.
732
00:46:11 --> 00:46:12
OK.
733
00:46:12 --> 00:46:13
There was another question.
734
00:46:13 --> 00:46:13
Yeah.
735
00:46:13 --> 00:46:18
AUDIENCE: [INAUDIBLE]
736
00:46:18 --> 00:46:19
PROFESSOR STRANG:
Of number four?
737
00:46:19 --> 00:46:22
Which number four in which?
738
00:46:22 --> 00:46:23
Oh, in the quiz.
739
00:46:23 --> 00:46:25
Oh yes, right.
740
00:46:25 --> 00:46:27
Yes.
741
00:46:27 --> 00:46:28
Did I tell you what
problem four was?
742
00:46:28 --> 00:46:29
No.
743
00:46:29 --> 00:46:31
I hope not.
744
00:46:31 --> 00:46:39
OK problem four in the quiz.
745
00:46:39 --> 00:46:41
It's about a delta function?
746
00:46:41 --> 00:46:43
Yeah.
747
00:46:43 --> 00:46:52
What do I know, what do
you want me to tell you?
748
00:46:52 --> 00:46:57
So the delta, of course, comes
in as we've seen it, as the
749
00:46:57 --> 00:47:01
right hand side of a
differential equation.
750
00:47:01 --> 00:47:05
So it might be the right hand
side even of this equation.
751
00:47:05 --> 00:47:12
So if this equation was
delta(x), or x-1/2
752
00:47:12 --> 00:47:20
or something.
753
00:47:20 --> 00:47:24
I mean, the essential thing is
that when delta's on the right
754
00:47:24 --> 00:47:32
side, that gives you
a drop in the slope.
755
00:47:32 --> 00:47:35
Suppose I just have a first
order equation like d.
756
00:47:35 --> 00:47:35
I'll call it dz/dx=delta(x-a).
757
00:47:35 --> 00:47:41
758
00:47:41 --> 00:47:45
Yeah.
759
00:47:45 --> 00:47:53
And suppose that I know
that z(0) starts at zero.
760
00:47:53 --> 00:47:58
You've got to be able to solve
that equation, so this is
761
00:47:58 --> 00:48:02
a useful prep for that.
762
00:48:02 --> 00:48:08
That would be a good equation
to know the solution to.
763
00:48:08 --> 00:48:13
And what kind of
function is this?
764
00:48:13 --> 00:48:17
What kind of a function
is z(x) there?
765
00:48:17 --> 00:48:20
I just use the letter z
to have a new letter.
766
00:48:20 --> 00:48:28
z(x) will be a step.
767
00:48:28 --> 00:48:33
Right. z(x) will be a
step function, yes.
768
00:48:33 --> 00:48:36
That's right.
769
00:48:36 --> 00:48:41
OK, so the solution is that
at the point a, which I'm
770
00:48:41 --> 00:48:44
presuming is beyond zero, I
come along at zero
771
00:48:44 --> 00:48:45
and I step up.
772
00:48:45 --> 00:48:45
Yep.
773
00:48:45 --> 00:48:46
OK.
774
00:48:46 --> 00:48:48
That would be a picture
of z(x), yeah.
775
00:48:48 --> 00:48:54
So it's basic delta
function material that
776
00:48:54 --> 00:49:04
I'm speaking about here.
777
00:49:04 --> 00:49:10
So z jumps by one and z is a
slope, then the slope jumps by
778
00:49:10 --> 00:49:15
one or drops by one, depending
on a plus or a minus sign here.
779
00:49:15 --> 00:49:19
The things that we've used to
deal with delta functions, so
780
00:49:19 --> 00:49:23
that's what Question
4b is about.
781
00:49:23 --> 00:49:27
The drop in slope, or the
jumps, or whatever happens
782
00:49:27 --> 00:49:31
when a delta function shows
up on the right side.
783
00:49:31 --> 00:49:33
Good question.
784
00:49:33 --> 00:49:33
Yep.
785
00:49:33 --> 00:49:36
AUDIENCE: [INAUDIBLE]
786
00:49:36 --> 00:49:38
PROFESSOR STRANG: 1.1.27.
787
00:49:38 --> 00:49:38
Well.
788
00:49:38 --> 00:49:41
AUDIENCE: [INAUDIBLE]
789
00:49:41 --> 00:49:44
PROFESSOR STRANG: Oh,
and then left a typo.
790
00:49:44 --> 00:49:53
AUDIENCE: [INAUDIBLE]
791
00:49:53 --> 00:49:54
PROFESSOR STRANG: Oh.
792
00:49:54 --> 00:49:58
I'm sorry, OK.
793
00:49:58 --> 00:50:05
1.1.27 My copy
isn't showing it.
794
00:50:05 --> 00:50:07
Yeah.
795
00:50:07 --> 00:50:10
I may have to punt
on that question.
796
00:50:10 --> 00:50:12
Or do you want me
to look at it?
797
00:50:12 --> 00:50:18
OK, can you maybe just pass
that the book up, alright.
798
00:50:18 --> 00:50:21
I'll try to read what
that question was.
799
00:50:21 --> 00:50:23
OK.
800
00:50:23 --> 00:50:27
Yeah, maybe this is a
question to answer.
801
00:50:27 --> 00:50:33
This is probably the one new
question that got added.
802
00:50:33 --> 00:50:39
OK, yeah.
803
00:50:39 --> 00:50:41
Fair enough.
804
00:50:41 --> 00:50:46
So this is continuing the
discussion that you asked me
805
00:50:46 --> 00:50:50
to start here about a, the
first difference matrix, OK.
806
00:50:50 --> 00:50:57
So I'll go a little
more over that.
807
00:50:57 --> 00:51:07
So in the book here, this
writes down a matrix A_0
808
00:51:07 --> 00:51:10
I'll discuss this matrix.
809
00:51:10 --> 00:51:16
So there's a difference matrix.
810
00:51:16 --> 00:51:20
You see what I mean by a
difference matrix, if were to
811
00:51:20 --> 00:51:26
multiply it by u, or something, I
812
00:51:26 --> 00:51:28
would get differences, right?
813
00:51:28 --> 00:51:33
I'd get u_1-u_0,
u_2-u_1, and u_3-u_2.
814
00:51:33 --> 00:51:36
815
00:51:36 --> 00:51:37
Good.
816
00:51:37 --> 00:51:39
So that's A_0 times u.
817
00:51:39 --> 00:51:40
Alright.
818
00:51:40 --> 00:51:45
So that's a difference matrix.
819
00:51:45 --> 00:51:47
What graph would
that come from?
820
00:51:47 --> 00:51:52
That's also the incidence
matrix of a very simple graph.
821
00:51:52 --> 00:51:56
This is connecting Chapter
1 with Chapter 2.
822
00:51:56 --> 00:51:59
It would be a line of springs,
it would be a graph.
823
00:51:59 --> 00:52:04
It's got edges and nodes.
it's got three edges,
824
00:52:04 --> 00:52:05
so I've got three rows.
825
00:52:05 --> 00:52:13
It's got four nodes so
I've got four columns.
826
00:52:13 --> 00:52:16
Are the columns
independent here?
827
00:52:16 --> 00:52:18
No, they never are.
828
00:52:18 --> 00:52:23
The vector of all ones would
have differences of all zeroes.
829
00:52:23 --> 00:52:26
So what would that, that
would be the difference
830
00:52:26 --> 00:52:31
matrix for fixed?
831
00:52:31 --> 00:52:33
Free?
832
00:52:33 --> 00:52:37
Fixed, free, circular what
would that difference
833
00:52:37 --> 00:52:41
matrix correspond to?
834
00:52:41 --> 00:52:42
Everybody's saying it.
835
00:52:42 --> 00:52:44
Say it a little louder just to.
836
00:52:44 --> 00:52:46
Free free.
837
00:52:46 --> 00:52:48
That's a free-free problem,
because they're all in there.
838
00:52:48 --> 00:52:50
We haven't knocked any out.
839
00:52:50 --> 00:52:52
There are no boundary
conditions yet.
840
00:52:52 --> 00:52:57
That's a free free, so that
A_0 would be free free.
841
00:52:57 --> 00:53:03
OK.
842
00:53:03 --> 00:53:06
I'll take one more case and
then I think we're at time.
843
00:53:06 --> 00:53:10
Suppose it was fixed fixed?
844
00:53:10 --> 00:53:14
What would be the difference
matrix that would correspond
845
00:53:14 --> 00:53:18
to, how would I change that
matrix if my problem
846
00:53:18 --> 00:53:20
became fixed fixed?
847
00:53:20 --> 00:53:26
So now I'm fixing that u, I'm
fixing that u, in the mass
848
00:53:26 --> 00:53:31
spring case I'm adding
supports there.
849
00:53:31 --> 00:53:34
How would that
change the matrix?
850
00:53:34 --> 00:53:36
First and fourth, good.
851
00:53:36 --> 00:53:37
Say it again?
852
00:53:37 --> 00:53:43
First and fourth
columns would go.
853
00:53:43 --> 00:53:48
So fixed free would
then be three by two.
854
00:53:48 --> 00:53:52
Free free was three by four.
855
00:53:52 --> 00:53:55
Yeah, I'm glad this question
came up because this is the
856
00:53:55 --> 00:53:59
right thing for you to
be thinking about in
857
00:53:59 --> 00:54:04
connection with the recent
question you asked.
858
00:54:04 --> 00:54:05
OK.
859
00:54:05 --> 00:54:08
Maybe that's the right place to
stop, because now you've asked
860
00:54:08 --> 00:54:13
questions that are really on
target for what we've done,
861
00:54:13 --> 00:54:16
and I hope useful to you.
862
00:54:16 --> 00:54:23
OK, see you guys tomorrow
evening at 7:30 in 54-100, OK.
863
00:54:23 --> 00:54:25