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Okay, this is linear algebra, lecture four.
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And, the first thing I have to do is something
that was on the list for last time, but here
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00:00:24,810 --> 00:00:26,169
it is now.
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00:00:26,169 --> 00:00:29,369
What's the inverse of a product?
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00:00:29,369 --> 00:00:36,300
If I multiply two matrices together and I
know their inverses, how do I get the inverse
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00:00:36,300 --> 00:00:39,559
of A times B?
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00:00:39,559 --> 00:00:46,760
So I know what inverses mean for a single
matrix A and for a matrix B.
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00:00:46,760 --> 00:00:53,229
What matrix do I multiply by to get the identity
if I have A B?
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00:00:53,229 --> 00:00:57,100
Okay, that'll be simple but so basic.
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Then I'm going to use that to -- I will have
a product of matrices and the product that
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00:01:06,120 --> 00:01:13,659
we'll meet will be these elimination matrices
and the net result of today's lectures is
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the big formula for elimination, so the net
result of today's lecture is this great way
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to look at Gaussian elimination.
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We know that we get from A to U by elimination.
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We know the steps -- but now we get the right
way to look at it, A equals L U.
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So that's the high point for today.
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Okay.
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Can I take the easy part, the first step first?
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So, suppose A is invertible -- and of course
it's going to be a big question, when is the
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matrix invertible?
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But let's say A is invertible and B is invertible,
then what matrix gives me the inverse of A B?
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So that's the direct question.
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What's the inverse of A B?
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00:02:08,869 --> 00:02:12,120
Do I multiply those separate inverses?
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Yes. I multiply the two matrices A inverse
and B inverse, but what order do I multiply?
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In reverse order.
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And you see why.
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So the right thing to put here is B inverse
A inverse.
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That's the inverse I'm after.
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00:02:31,820 --> 00:02:38,940
We can just check that A B times that matrix
gives the identity.
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00:02:38,950 --> 00:02:39,950
Okay.
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00:02:39,950 --> 00:02:46,480
So why -- once again, it's this fact that
I can move parentheses around.
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00:02:46,480 --> 00:02:52,320
I can just erase them all and do the multiplications
any way I want to.
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So what's the right multiplication to do first?
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B times B inverse.
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This product here I is the identity.
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Then A times the identity is the identity
and then finally A times A inverse gives the
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00:03:08,100 --> 00:03:10,060
identity.
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00:03:10,060 --> 00:03:15,680
So forgive the dumb example in the book.
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00:03:15,680 --> 00:03:21,180
Why do you, do the inverse things in reverse
order?
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00:03:21,189 --> 00:03:27,019
It's just like -- you take off your shoes,
you take off your socks, then the good way
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00:03:27,020 --> 00:03:35,060
to invert that process is socks back on first,
then shoes.
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00:03:35,060 --> 00:03:36,660
Sorry, okay.
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00:03:36,660 --> 00:03:41,320
I'm sorry that's on the tape.
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00:03:41,320 --> 00:03:46,460
And, of course, on the other side we should
really just check -- on the other side I have
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00:03:46,460 --> 00:03:47,480
B inverse,
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00:03:47,480 --> 00:03:54,099
A inverse. That does multiply A B, and this
time it's these guys that give the identity,
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00:03:54,099 --> 00:03:58,040
squeeze down, they give the identity, we're
in shape.
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00:03:58,040 --> 00:04:01,700
Okay. So there's the inverse.
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00:04:01,700 --> 00:04:08,480
Good. While we're at it, let me do a transpose,
because the next lecture has got a lot to
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-- involves transposes.
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00:04:09,780 --> 00:04:16,860
So how do I -- if I transpose a matrix, I'm
talking about square, invertible matrices
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00:04:16,860 --> 00:04:18,298
right now.
54
00:04:18,300 --> 00:04:23,500
If I transpose one, what's its inverse?
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00:04:23,500 --> 00:04:27,640
Well, the nice formula is -- let's see.
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00:04:27,640 --> 00:04:34,760
Let me start from A, A inverse equal the identity.
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And let me transpose both sides.
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That will bring a transpose into the picture.
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00:04:43,199 --> 00:04:48,300
So if I transpose the identity matrix, what
do I have?
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The identity, right?
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00:04:49,600 --> 00:04:54,990
If I exchange rows and columns, the identity
is a symmetric matrix.
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It doesn't know the difference.
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00:04:56,940 --> 00:05:07,780
If I transpose these guys, that product, then
again it turns out that I have to reverse
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00:05:07,780 --> 00:05:09,160
the order.
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00:05:09,160 --> 00:05:15,260
I can transpose them separately, but when
I multiply, those transposes come in the opposite
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00:05:15,260 --> 00:05:16,220
order.
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00:05:16,220 --> 00:05:23,600
So it's A inverse transpose times A transpose
giving the identity.
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00:05:23,600 --> 00:05:26,700
So that's -- this equation is -- just comes
directly from that
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00:05:26,700 --> 00:05:32,690
one. But this equation tells me what I wanted
to know, namely what is the inverse of this
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00:05:32,690 --> 00:05:35,700
guy A transpose?
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00:05:35,700 --> 00:05:43,180
What's the inverse of that -- if I transpose
a matrix, what'ss the inverse of the result?
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00:05:43,180 --> 00:05:47,000
And this equation tells me that here it is.
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This is the inverse of A transpose.
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Inverse of A transpose.
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Of A transpose.
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00:06:08,100 --> 00:06:13,480
So I'll put a big circle around that, because
that's the answer, that's the best answer
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00:06:13,480 --> 00:06:16,560
we could hope for.
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00:06:16,560 --> 00:06:24,640
That if you want to know the inverse of A
transpose and you know the inverse of A, then
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00:06:24,640 --> 00:06:26,640
you just transpose that.
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00:06:26,640 --> 00:06:33,790
So in a -- to put it another way, transposing
and inversing you can do in either order for
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00:06:33,790 --> 00:06:35,390
a single matrix.
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00:06:35,390 --> 00:06:42,770
Okay. So these are like basic facts that we
can now use, all right -- so now I put it
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00:06:42,770 --> 00:06:44,920
to use.
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00:06:44,920 --> 00:06:52,980
I put it to use by thinking -- we're really
completing, the subject of elimination.
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00:06:52,980 --> 00:07:05,920
Actually, -- the thing about elimination is
it's the right way to understand what the
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00:07:05,930 --> 00:07:06,930
matrix has got.
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00:07:06,930 --> 00:07:15,550
This A equal L U is the most basic factorization
of a matrix.
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00:07:15,550 --> 00:07:23,960
I always worry that you will think this course
is all elimination.
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It's just row operations.
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00:07:27,100 --> 00:07:29,420
And please don't.
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00:07:29,420 --> 00:07:41,120
We'll be beyond that, but it's the right algebra
to do first.
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00:07:41,120 --> 00:07:48,630
Okay. So, now I'm coming near the end of it,
but I want to get it in a decent form.
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00:07:48,630 --> 00:07:52,270
So my decent form is matrix form.
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00:07:52,270 --> 00:07:59,360
I have a matrix A, let's suppose it's a good
matrix, I can do elimination, no row exchanges
95
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-- So no row exchanges for now.
96
00:08:03,220 --> 00:08:07,640
Pivots all fine, nothing zero in the pivot
position.
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00:08:07,640 --> 00:08:10,240
I get to the very end, which is U.
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00:08:10,240 --> 00:08:12,000
So I get from A to U.
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00:08:12,000 --> 00:08:15,220
And I want to know what's the connection?
100
00:08:15,220 --> 00:08:17,680
How is A related to U?
101
00:08:17,690 --> 00:08:21,991
And this is going to tell me that there's
a matrix L that connects them.
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Okay.
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00:08:23,000 --> 00:08:28,700
Can I do it for a two by two first?
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Okay. Two by two, elimination.
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00:08:32,679 --> 00:08:37,179
Okay, so I'll do it under here.
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00:08:37,186 --> 00:08:46,680
Okay. So let my matrix A be -- We'll keep
it simple, say two and an eight, so we know
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that the first pivot is a two, and the multiplier's
going to be a four and then let me put a one
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00:08:54,029 --> 00:08:58,720
here and what number do I not want to put
there?
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00:08:58,720 --> 00:09:05,780
Four. I don't want a four there, because in
that case, the second pivot would not -- we
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00:09:05,780 --> 00:09:07,642
wouldn't have a second pivot.
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00:09:07,642 --> 00:09:11,360
The matrix would be singular, general screw-up.
Okay.
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00:09:11,360 --> 00:09:15,090
So let me put some other number here like
seven.
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Okay.
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00:09:16,580 --> 00:09:24,680
Okay. Now I want to operate on that with my
elementary matrix.
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00:09:24,680 --> 00:09:28,340
So what's the elementary matrix?
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Strictly speaking, it's E21, because it's
the guy that's going to produce a zero in
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that position.
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And it's going to produce U in one shot, because
it's just a two by two matrix.
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So two one and I'm going to take four of those
away from those, produce that zero and leave
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a three there.
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And that's U.
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00:09:53,080 --> 00:09:55,140
And what's the matrix that did it?
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00:09:55,140 --> 00:09:56,900
Quick review, then.
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What's the elimination elementary matrix E21
-- it's one zero, thanks.
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00:10:03,060 --> 00:10:05,760
And -- negative four one,
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00:10:05,760 --> 00:10:07,880
right. Good.
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00:10:07,880 --> 00:10:14,100
Okay. So that -- you see the difference between
this and what I'm shooting for.
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00:10:14,100 --> 00:10:21,980
I'm shooting for A on one side and the other
matrices on the other side of the equation.
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Okay.
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00:10:22,990 --> 00:10:25,940
So I can do that right away.
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00:10:25,940 --> 00:10:29,500
Now here's going to be my A equals L U.
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00:10:29,500 --> 00:10:38,160
And you won't have any trouble telling me
what -- so A is still two one eight seven.
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00:10:38,160 --> 00:10:44,940
L is what you're going to tell me and U is
still two one zero three.
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Okay. So what's L in this case?
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00:10:49,800 --> 00:10:57,220
Well, first -- so how is L related to this
E guy?
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00:10:57,220 --> 00:11:02,650
It's the inverse, because I want to multiply
through by the inverse of this, which will
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00:11:02,650 --> 00:11:08,570
put the identity here, and the inverse will
show up there and I'll call it L.
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00:11:08,570 --> 00:11:11,360
So what is the inverse of this?
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00:11:11,360 --> 00:11:17,640
Remember those elimination matrices are easy
to invert.
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00:11:17,650 --> 00:11:30,130
The inverse matrix for this one is 1 0 4 1,
it has the plus sign because it adds back
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what this removes.
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00:11:32,200 --> 00:11:39,860
Okay. Do you want -- if we did the numbers
right, we must -- this should be correct.
143
00:11:39,860 --> 00:11:42,380
Okay. And of course it is.
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00:11:42,390 --> 00:11:47,730
That says the first row's right, four times
the first row plus the second row is eight
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00:11:47,730 --> 00:11:49,700
seven. Good. Okay.
146
00:11:49,720 --> 00:11:51,920
That's simple, two by two.
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00:11:51,920 --> 00:11:57,380
But it already shows the form that we're headed
for.
148
00:11:57,390 --> 00:12:01,190
It shows -- so what's the L stand for?
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00:12:01,190 --> 00:12:02,640
Why the letter L?
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00:12:02,640 --> 00:12:09,860
If U stood for upper triangular, then of course
L stands for lower triangular.
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00:12:09,860 --> 00:12:16,920
And actually, it has ones on the diagonal,
where this thing has the pivots on the diagonal.
152
00:12:16,920 --> 00:12:27,080
Oh, sometimes we may want to separate out
the pivots, so can I just mention that sometimes
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00:12:27,089 --> 00:12:34,339
we could also write this as -- we could have
this one zero four one -- I'll just show you
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00:12:34,340 --> 00:12:39,160
how I would divide out this matrix of pivots
-- two three.
155
00:12:39,160 --> 00:12:41,880
There's a diagonal matrix.
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00:12:41,880 --> 00:12:45,980
And I just -- whatever is left is here.
157
00:12:45,980 --> 00:12:49,620
Now what's left?
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00:12:49,630 --> 00:12:57,140
If I divide this first row by two to pull
out the two, then I have a one and a one half.
159
00:12:57,140 --> 00:13:02,580
And if I divide the second row by three to
pull out the three, then I have a one.
160
00:13:02,580 --> 00:13:11,180
So if this is L
U, this is maybe called L D or pivot U.
161
00:13:11,180 --> 00:13:21,100
And now it's a little more balanced, because
we have ones on the diagonal here and here.
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00:13:21,100 --> 00:13:22,920
And the diagonal matrix in the middle.
163
00:13:22,920 --> 00:13:24,380
So both of those...
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00:13:24,380 --> 00:13:29,400
Matlab would produce either one.
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00:13:29,400 --> 00:13:32,860
I'll basically stay with L U.
166
00:13:32,860 --> 00:13:39,960
Okay. Now I have to think about bigger than
two by two.
167
00:13:39,960 --> 00:13:43,780
But right now, this was just like easy exercise.
168
00:13:43,780 --> 00:13:50,920
And, to tell the truth, this one was a minus
sign and this one was a plus sign.
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00:13:50,920 --> 00:13:54,740
I mean, that's the only difference.
170
00:13:54,740 --> 00:14:02,700
But, with three by three, there's a more significant
difference.
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00:14:02,700 --> 00:14:05,740
Let me show you how that works.
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00:14:05,740 --> 00:14:17,760
Let me move up to a three by three, let's
say some matrix A, okay?
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00:14:17,760 --> 00:14:20,120
Let's imagine it's three by three.
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00:14:20,130 --> 00:14:23,050
I won't write numbers down for now.
175
00:14:23,050 --> 00:14:29,351
So what's the first elimination step that
I do, the first matrix I multiply it by, what
176
00:14:29,351 --> 00:14:30,870
letter will I use for
177
00:14:30,870 --> 00:14:41,490
that? It'll be E two one, because it's -- the
first step will be to get a zero in that two
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00:14:41,490 --> 00:14:43,520
one position. right?
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00:14:43,520 --> 00:14:48,260
And then the next step will be to get a zero
in the three one position.
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00:14:48,260 --> 00:14:53,560
And the final step will be to get a zero in
the three two
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00:14:53,560 --> 00:14:57,089
That's what elimination is, and it produced
U. position.
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00:14:57,089 --> 00:15:06,820
And again, no row exchanges.
183
00:15:06,820 --> 00:15:13,180
I'm taking the nice case, now, the typical
case, too -- when I don't have to do any row
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00:15:13,180 --> 00:15:15,920
exchange, all I do is these elimination steps.
185
00:15:15,920 --> 00:15:16,900
Okay.
186
00:15:16,900 --> 00:15:22,880
Now, suppose I want that stuff over on the
right-hand side, as I really do.
187
00:15:22,880 --> 00:15:25,470
That's, like, my point here.
188
00:15:25,470 --> 00:15:30,300
I can multiply these together to get a matrix
E, but I want it over on the right.
189
00:15:30,300 --> 00:15:33,670
I want its inverse over there.
190
00:15:33,670 --> 00:15:39,160
So what's the right expression now?
191
00:15:39,160 --> 00:15:46,440
If I write A and U, what goes there?
192
00:15:46,440 --> 00:15:47,060
Okay.
193
00:15:47,060 --> 00:15:50,040
So I've got the inverse of this, I've got
three matrices in
194
00:15:50,040 --> 00:15:52,020
a row now.
195
00:15:52,020 --> 00:15:58,300
And it's their inverses that are going to
show up, because each one is easy to invert.
196
00:15:58,300 --> 00:16:01,460
Question is, what about the whole bunch?
197
00:16:01,460 --> 00:16:04,780
How easy is it to invert the whole bunch?
198
00:16:04,780 --> 00:16:08,630
So, that's what we know how to do.
199
00:16:08,630 --> 00:16:12,840
We know how to invert, we should take the
separate inverses, but they go in the opposite
200
00:16:12,840 --> 00:16:13,840
order.
201
00:16:13,840 --> 00:16:15,800
So what goes here?
202
00:16:15,800 --> 00:16:21,640
E three two inverse, right, because I'll multiply
from the left by E three two inverse, then
203
00:16:21,640 --> 00:16:26,600
I'll pop it up next to U.
204
00:16:26,600 --> 00:16:30,880
And then will come E three one inverse.
205
00:16:30,880 --> 00:16:40,900
And then this'll be the only guy left standing
and that's gone when I do an E two one inverse.
206
00:16:40,910 --> 00:16:43,590
So there is L.
207
00:16:43,590 --> 00:16:47,300
That's L U.
208
00:16:47,300 --> 00:16:50,860
L is product of inverses.
209
00:16:50,860 --> 00:16:57,680
Now you still can ask why is this guy preferring
inverses?
210
00:16:57,680 --> 00:16:59,570
And let me explain why.
211
00:16:59,570 --> 00:17:05,619
Let me explain why is this product nicer than
this one?
212
00:17:05,619 --> 00:17:10,380
This product turns out to be better than this
one.
213
00:17:10,380 --> 00:17:15,060
Let me take a typical case here.
214
00:17:15,060 --> 00:17:17,250
Let me take a typical case.
215
00:17:17,250 --> 00:17:23,000
So let me -- I have to do three by three for
you to see the improvement.
216
00:17:23,000 --> 00:17:27,400
Two by two, it was just one E, no problem.
217
00:17:27,400 --> 00:17:29,960
But let me go up to this case.
218
00:17:29,960 --> 00:17:41,460
Suppose my matrices E21 -- suppose E21 has
a minus two in there.
219
00:17:41,460 --> 00:17:49,840
Suppose that -- and now suppose -- oh, I'll
even suppose E31 is the identity.
220
00:17:49,840 --> 00:17:54,360
I'm going to make the point with just a couple
of these.
221
00:17:54,360 --> 00:17:54,900
Okay.
222
00:17:54,900 --> 00:18:02,720
Now this guy will have -- do something -- now
let's suppose minus five one.
223
00:18:02,720 --> 00:18:07,560
Okay. There's typical.
224
00:18:07,560 --> 00:18:12,760
That's a typical case in which we didn't need
an E31. Maybe we already had a zero in that
225
00:18:12,760 --> 00:18:14,660
three one position.
226
00:18:14,660 --> 00:18:17,560
Okay.
227
00:18:17,560 --> 00:18:24,700
Let me see -- is that going to be enough to,
show my point?
228
00:18:24,700 --> 00:18:29,500
Let me do that multiplication.
229
00:18:29,500 --> 00:18:32,640
So if I do that multiplication it's like good
practice to
230
00:18:32,640 --> 00:18:35,000
multiply these matrices.
231
00:18:35,000 --> 00:18:39,900
Tell me what's above the diagonal when I do
this multiplication?
232
00:18:39,900 --> 00:18:44,500
All zeroes. When I do this multiplication,
I'm going to get ones on the diagonal and
233
00:18:44,500 --> 00:18:48,280
zeroes above.
234
00:18:48,280 --> 00:18:50,140
Because -- what does that say?
235
00:18:50,140 --> 00:18:54,540
That says that I'm subtracting rows from lower
rows.
236
00:18:54,549 --> 00:19:00,970
So nothing is moving upwards as it did last
time in Gauss-Jordan. Okay.
237
00:19:00,970 --> 00:19:10,280
Now -- so really, what I have to do is check
this minus two one zero, now this is -- what's
238
00:19:10,280 --> 00:19:11,240
that number?
239
00:19:11,240 --> 00:19:15,740
This is the number that I'm really have in
mind.
240
00:19:15,740 --> 00:19:19,900
That number is ten.
241
00:19:19,900 --> 00:19:25,620
And this one is -- what goes here?
242
00:19:25,620 --> 00:19:28,960
Row three against column two, it looks like
the minus five.
243
00:19:32,000 --> 00:19:35,100
What – it's that ten.
244
00:19:35,100 --> 00:19:37,020
How did that ten get in there?
245
00:19:37,020 --> 00:19:39,040
I don't like that ten.
246
00:19:39,040 --> 00:19:42,220
I mean -- of course, I don't want to erase
it, because it's right.
247
00:19:42,220 --> 00:19:45,840
But I don't want it there.
248
00:19:45,840 --> 00:19:53,800
It's because -- the ten got in there because
I subtracted two of row one from row two,
249
00:19:53,800 --> 00:19:58,760
and then I subtracted five of that new row
two from row three.
250
00:19:58,760 --> 00:20:04,920
So doing it in that order, how did row one
effect row three?
251
00:20:04,920 --> 00:20:10,480
Well, it did, because two of it got removed
from row two and then five of those got removed
252
00:20:10,480 --> 00:20:11,590
from row three.
253
00:20:11,590 --> 00:20:17,800
So altogether ten of row one got thrown into
row three.
254
00:20:17,800 --> 00:20:27,540
Now my point is in the reverse direction -- so
now I can do it -- below it I'll do the inverses.
255
00:20:27,540 --> 00:20:28,100
Okay.
256
00:20:28,100 --> 00:20:29,820
And, of course, opposite order.
257
00:20:29,820 --> 00:20:31,660
Reverse order.
258
00:20:31,660 --> 00:20:35,960
Reverse order.
259
00:20:35,960 --> 00:20:44,020
Okay. So now this is going to -- this is the
E that goes on the left side.
260
00:20:44,020 --> 00:20:47,790
Left of A.
261
00:20:47,790 --> 00:20:54,340
Now I'm going to do the inverses in the opposite
order, so what's the -- So the opposite order
262
00:20:54,340 --> 00:20:57,530
means I put this inverse first.
263
00:20:57,530 --> 00:20:58,890
And what is its inverse?
264
00:20:58,890 --> 00:21:04,660
What's the inverse of E21? Same thing with
a plus sign, right?
265
00:21:04,660 --> 00:21:12,730
For the individual matrices, instead of taking
away two I add back two of row one to row
266
00:21:12,730 --> 00:21:16,620
two, so no problem.
267
00:21:16,620 --> 00:21:21,400
And now, in reverse order, I want to invert
that.
268
00:21:21,400 --> 00:21:22,840
Just right?
269
00:21:22,840 --> 00:21:25,540
I'm doing just this, this.
270
00:21:25,540 --> 00:21:36,320
So now the inverse is again the same thing,
but add in the five.
271
00:21:36,320 --> 00:21:45,460
And now I'll do that multiplication and I'll
get a happy result.
272
00:21:45,460 --> 00:21:47,800
I hope.
273
00:21:47,800 --> 00:21:50,220
Did I do it right so far?
274
00:21:50,220 --> 00:21:50,960
Yes, okay.
275
00:21:50,960 --> 00:21:51,640
Let me do the multiplication.
276
00:21:51,650 --> 00:21:53,350
I believe this comes out.
277
00:21:53,350 --> 00:21:56,660
So row one of the answer is one zero zero.
278
00:21:56,660 --> 00:21:59,300
Oh, I know that all this is going to be left,
279
00:21:59,300 --> 00:22:04,080
right? Then I have two one zero.
280
00:22:04,090 --> 00:22:07,559
So I get two one zero there, right?
281
00:22:07,560 --> 00:22:10,060
And what's the third row?
282
00:22:10,060 --> 00:22:15,160
What's the third row in this product?
283
00:22:15,160 --> 00:22:18,620
Just read it out to me, the third row?
284
00:22:18,640 --> 00:22:22,980
0 5 1
285
00:22:22,980 --> 00:22:27,240
Because one way to say is –
this is saying take one of the
286
00:22:27,240 --> 00:22:29,640
last row and there it is.
287
00:22:29,640 --> 00:22:32,580
And this is my matrix L.
288
00:22:32,590 --> 00:22:38,450
And it's the one that goes on the left of
U.
289
00:22:38,450 --> 00:22:45,190
It goes into -- what do I mean here?
290
00:22:45,190 --> 00:22:52,040
Maybe rather than saying left of A, left of
U, let me right down again what I mean.
291
00:22:52,040 --> 00:22:58,600
E A is U, whereas A is L U.
292
00:22:58,600 --> 00:23:01,560
Okay.
293
00:23:01,560 --> 00:23:06,280
Let me make the point now in words.
294
00:23:06,300 --> 00:23:11,090
The order that the matrices come for L is
the right order.
295
00:23:11,090 --> 00:23:21,780
The two and the five don't sort of interfere
to produce this ten one. In the
296
00:23:21,800 --> 00:23:26,919
right order, the multipliers just sit in the
matrix L.
297
00:23:26,919 --> 00:23:35,530
That's the point -- that if I want to know
L, I have no work to do.
298
00:23:35,530 --> 00:23:41,370
I just keep a record of what those multipliers
were, and that gives me L.
299
00:23:41,370 --> 00:23:52,610
So I'll draw the -- let me say it.
300
00:23:52,610 --> 00:23:56,320
So this is the A=L U.
301
00:23:56,320 --> 00:24:14,010
So if no row exchanges, the multipliers that
those numbers that we multiplied rows by and
302
00:24:14,010 --> 00:24:27,550
subtracted, when we did an elimination step
-- the multipliers go directly into L.
303
00:24:27,550 --> 00:24:29,950
Okay.
304
00:24:29,950 --> 00:24:42,770
So L is -- this is the way, to look at elimination.
305
00:24:42,770 --> 00:24:51,510
You go through the elimination steps, and
actually if you do it right, you can throw
306
00:24:51,510 --> 00:24:57,120
away A as you create L U.
307
00:24:57,120 --> 00:25:08,520
If you think about it, those steps of elimination,
as when you've finished with row two of A,
308
00:25:08,520 --> 00:25:16,540
you've created a new row two of U, which you
have to save, and you've created the multipliers
309
00:25:16,540 --> 00:25:22,980
that you used -- which you have to save,
and then you can forget A.
310
00:25:22,980 --> 00:25:25,919
So because it's all there in L and U.
311
00:25:25,919 --> 00:25:41,580
So that's -- this moment is maybe the new
insight in elimination that comes from matrix
312
00:25:41,580 --> 00:25:44,070
-- doing it in matrix form.
313
00:25:44,070 --> 00:25:52,350
So it was -- the product of Es is -- we can't
see what that product of Es is.
314
00:25:52,350 --> 00:25:56,350
The matrix E is not a particularly attractive
one.
315
00:25:56,350 --> 00:26:02,130
What's great is when we put them on the other
side -- their inverses in the opposite order,
316
00:26:02,130 --> 00:26:06,210
there the L comes out just right. Okay.
317
00:26:06,210 --> 00:26:09,580
Now -- oh gosh, so today's a sort of,
318
00:26:09,580 --> 00:26:14,160
like, practical day.
319
00:26:14,160 --> 00:26:20,670
Can we think together how expensive is elimination?
320
00:26:20,670 --> 00:26:24,570
How many operations do we do?
321
00:26:24,570 --> 00:26:32,520
So this is now a kind of new topic which I
didn't list as -- on the program, but here
322
00:26:32,540 --> 00:26:34,980
it came. Here it comes.
323
00:26:34,980 --> 00:26:55,780
How many operations on an n by n matrix A.
324
00:26:55,780 --> 00:26:58,020
I mean, it's a very practical question.
325
00:26:58,020 --> 00:27:08,280
Can we solve systems of order a thousand,
in a second or a minute or a week?
326
00:27:08,280 --> 00:27:16,460
Can we solve systems of order a million in
a second or an hour or a week?
327
00:27:16,460 --> 00:27:25,059
I mean, what's the -- if it's n by n, we often
want to take n bigger.
328
00:27:25,059 --> 00:27:28,700
I mean, we've put in more information.
329
00:27:28,700 --> 00:27:33,880
We make the whole thing is more accurate for
the bigger matrix.
330
00:27:33,880 --> 00:27:39,000
But it's more expensive, too, and the question
is how much more expensive?
331
00:27:39,000 --> 00:27:41,679
If I have matrices of order a hundred.
332
00:27:41,679 --> 00:27:43,150
Let's say a hundred by a hundred.
333
00:27:43,150 --> 00:27:47,660
Let me take n to be a hundred.
334
00:27:47,660 --> 00:27:50,840
Say n equal a hundred.
335
00:27:50,840 --> 00:27:54,860
How many steps are we doing?
336
00:27:54,860 --> 00:28:02,770
How many operations are we actually doing
that we -- And let's suppose there aren't
337
00:28:02,770 --> 00:28:07,799
any zeroes, because of course if a matrix
has got a lot of zeroes in good places, we
338
00:28:07,800 --> 00:28:09,870
don't have to do those operations, and,
339
00:28:09,870 --> 00:28:13,500
it'll be much faster.
340
00:28:13,500 --> 00:28:21,940
But -- so just think for a moment about the
first step.
341
00:28:21,940 --> 00:28:27,179
So here's our matrix A, hundred by a hundred.
342
00:28:27,180 --> 00:28:34,880
And the first step will be -- that column,
is got zeroes down
343
00:28:34,880 --> 00:28:41,080
here. So it's down to 99 by 99, right?
344
00:28:41,080 --> 00:28:45,920
That's really like the first stage of elimination,
345
00:28:45,920 --> 00:28:48,480
to get from this hundred
346
00:28:48,490 --> 00:28:55,600
by hundred non-zero matrix to this stage where
the first pivot is sitting up here and the
347
00:28:55,600 --> 00:28:59,440
first row's okay the first column is okay.
348
00:28:59,440 --> 00:29:05,120
So, eventually -- how many steps did that
take?
349
00:29:05,120 --> 00:29:07,400
You see, I'm trying to get an idea.
350
00:29:07,409 --> 00:29:11,860
Is the answer proportional to n?
351
00:29:11,860 --> 00:29:17,260
Is the total number of steps in elimination,
the total number, is it proportional to n
352
00:29:17,260 --> 00:29:22,520
-- in which case if I double n from a hundred
to two hundred -- does it take me twice as
353
00:29:22,520 --> 00:29:24,140
long?
354
00:29:24,140 --> 00:29:27,340
Does it square, so it would take me four times
as long?
355
00:29:27,350 --> 00:29:30,600
Does it cube so it would take me eight times
as long?
356
00:29:30,600 --> 00:29:37,120
Or is it n factorial, so it would take me
a hundred times as long?
357
00:29:37,120 --> 00:29:41,560
I think, you know, from a practical point
of view, we have to have some idea of the
358
00:29:41,560 --> 00:29:44,280
cost, here.
359
00:29:44,280 --> 00:29:48,720
So these are the questions that I'm -- let
me ask those questions again.
360
00:29:48,720 --> 00:29:54,900
Is it proportional -- does it go like n, like
n squared, like n cubed -- or some higher
361
00:29:54,900 --> 00:29:56,420
power of n?
362
00:29:56,420 --> 00:30:04,580
Like n factorial where every step up multiplies
by a hundred and then by a hundred and one
363
00:30:04,580 --> 00:30:07,980
and then by a hundred and two -- which is
it?
364
00:30:07,980 --> 00:30:14,320
Okay, so that's the only way I know to answer
that is to think through what we actually
365
00:30:14,320 --> 00:30:16,100
had to do.
366
00:30:16,100 --> 00:30:17,100
Okay.
367
00:30:17,100 --> 00:30:22,680
So what was the cost here?
368
00:30:22,680 --> 00:30:23,300
Well, let's see.
369
00:30:23,309 --> 00:30:25,700
What do I mean by an operation?
370
00:30:25,700 --> 00:30:32,549
I guess I mean, well an addition or -- yeah.
371
00:30:32,549 --> 00:30:33,549
No big deal.
372
00:30:33,549 --> 00:30:38,980
I guess I mean an addition or a subtraction
or a multiplication or a division.
373
00:30:38,980 --> 00:30:41,540
Okay.
374
00:30:41,540 --> 00:30:48,980
And actually, what operation I doing all the
time?
375
00:30:48,980 --> 00:30:58,870
When I multiply row one by multiplier L and
I subtract from row six.
376
00:30:58,870 --> 00:31:01,060
What's happening there individually?
377
00:31:01,060 --> 00:31:03,460
What's going on?
378
00:31:03,460 --> 00:31:08,490
If I multiply -- I do a multiplication by
L and then a subtraction.
379
00:31:08,490 --> 00:31:15,610
So I guess operation -- Can I count that for
the moment as, like, one operation?
380
00:31:15,610 --> 00:31:18,520
Or you may want to count them separately.
381
00:31:18,520 --> 00:31:26,740
The typical operation is multiply plus a subtract.
382
00:31:26,740 --> 00:31:31,420
So if I count those together, my answer's
going to come out
383
00:31:31,420 --> 00:31:36,540
half as many as if -- I mean,
if I count them separately, I'd have a certain
384
00:31:36,540 --> 00:31:39,000
number of multiplies, certain number of subtracts.
385
00:31:39,000 --> 00:31:40,580
That's really want to do.
386
00:31:40,580 --> 00:31:43,360
Okay. How many have I got here?
387
00:31:43,360 --> 00:31:49,500
So, I think -- let's see.
388
00:31:49,500 --> 00:31:56,293
It's about -- well, how many, roughly?
389
00:31:56,293 --> 00:31:59,352
How many operations to get from here to here?
390
00:31:59,360 --> 00:32:05,800
Well, maybe one way to look at it is all these
numbers had to get changed.
391
00:32:05,800 --> 00:32:11,900
The first row didn't get changed, but all
the other rows got changed at this step.
392
00:32:11,900 --> 00:32:25,880
So this step -- well, I guess maybe -- shall
I say it cost about a hundred squared.
393
00:32:25,880 --> 00:32:32,100
I mean, if I had changed the first row, then
it would have been exactly hundred squared,
394
00:32:32,110 --> 00:32:35,549
because -- because that's how many numbers
are here.
395
00:32:35,549 --> 00:32:42,279
A hundred squared numbers is the total count
of the entry, and all but this insignificant
396
00:32:42,279 --> 00:32:44,299
first row got changed.
397
00:32:44,300 --> 00:32:47,660
So I would say about a hundred squared.
398
00:32:47,660 --> 00:32:48,620
Okay.
399
00:32:48,620 --> 00:32:54,150
Now, what about the next step?
400
00:32:54,150 --> 00:32:56,960
So now the first row is fine.
401
00:32:56,960 --> 00:33:00,020
The second row is fine.
402
00:33:00,020 --> 00:33:06,700
And I'm changing these zeroes are all fine,
so what's up with the second step?
403
00:33:06,700 --> 00:33:08,620
And then you're with me.
404
00:33:08,620 --> 00:33:10,380
Roughly, what's the cost?
405
00:33:10,390 --> 00:33:17,330
If this first step cost a hundred squared,
about, operations then this one, which is
406
00:33:17,330 --> 00:33:27,490
really working on this guy to produce this,
costs about what?
407
00:33:27,490 --> 00:33:31,250
How many operations to fix?
408
00:33:31,250 --> 00:33:36,610
About ninety-nine squared, or ninety-nine
times ninety-eight. But less, right?
409
00:33:36,610 --> 00:33:39,590
Less, because our problem's getting smaller.
410
00:33:39,590 --> 00:33:42,280
About ninety-nine squared.
411
00:33:42,280 --> 00:33:46,160
And then I go down and down and the next one
will be ninety-eight squared, the next ninety-seven
412
00:33:46,160 --> 00:33:53,750
squared and finally I'm down around one squared
or -- where it's just like the little numbers.
413
00:33:53,750 --> 00:33:55,890
The big numbers are here.
414
00:33:55,890 --> 00:34:06,190
So the number of operations is about n squared
plus that was n, right? n was a hundred?
415
00:34:06,190 --> 00:34:13,589
n squared for the first step, then n minus
one squared, then n minus two squared, finally
416
00:34:13,589 --> 00:34:23,040
down to three squared and two squared and
even one squared.
417
00:34:23,040 --> 00:34:27,929
No way I should have written that -- squeezed
that in.
418
00:34:27,929 --> 00:34:37,570
Let me try it so the count is n squared plus
n minus one squared plus -- all the way down
419
00:34:37,570 --> 00:34:41,980
to one squared.
420
00:34:41,980 --> 00:34:44,899
That's a pretty decent count.
421
00:34:44,899 --> 00:34:56,040
Admittedly, we didn't catch every single tiny
operation, but we got the right leading term
422
00:34:56,040 --> 00:34:57,040
here.
423
00:34:57,040 --> 00:35:01,180
And what do those add up to?
424
00:35:01,180 --> 00:35:10,030
Okay, so now we're coming to the punch of
this, question, this operation count.
425
00:35:10,030 --> 00:35:19,820
So the operations on the left side, on the
matrix A to finally get to U.
426
00:35:19,820 --> 00:35:28,540
And anybody -- so which of these quantities
is the right ballpark for that count?
427
00:35:28,540 --> 00:35:34,080
If I add a hundred squared to ninety nine
squared to ninety eight squared -- ninety
428
00:35:34,080 --> 00:35:44,020
seven squared, all the way down to two squared
then one squared, what have I got, about?
429
00:35:44,020 --> 00:35:48,100
It's just one of these -- let's identify it
first.
430
00:35:48,100 --> 00:35:49,020
Is it n?
431
00:35:49,020 --> 00:35:52,080
Certainly not.
432
00:35:52,080 --> 00:35:55,700
Is it n factorial?
433
00:35:55,700 --> 00:35:56,460
No.
434
00:35:56,460 --> 00:36:02,020
If it was n factorial, we would -- with determinants,
it is n factorial.
435
00:36:02,020 --> 00:36:12,400
I'll put in a bad mark against determinants,
because that -- okay, so what is it?
436
00:36:12,400 --> 00:36:18,580
It's n -- well, this is the answer.
437
00:36:18,580 --> 00:36:21,240
It's this order -- n cubed.
438
00:36:21,240 --> 00:36:26,380
It's like I have n terms, right?
439
00:36:26,380 --> 00:36:28,680
I've got n terms in this sum.
440
00:36:28,690 --> 00:36:31,140
And the biggest one is n squared.
441
00:36:31,140 --> 00:36:40,750
So the worst it could be would be n cubed,
but it's not as bad as -- it's n cubed times
442
00:36:40,750 --> 00:36:45,270
-- it's about one third of n cubed.
443
00:36:45,270 --> 00:36:53,690
That's the magic operation count.
444
00:36:53,690 --> 00:37:02,339
Somehow that one third takes account of the
fact that the numbers are getting smaller.
445
00:37:02,339 --> 00:37:06,900
If they weren't getting smaller, we would
have n terms times n squared, but it would
446
00:37:06,900 --> 00:37:08,240
be exactly n cubed.
447
00:37:08,240 --> 00:37:10,640
But our numbers are getting smaller -- actually,
row two and row one moves down to row three.
448
00:37:10,640 --> 00:37:19,400
do you remember where does one third come
in this -- I'll even allow a mention of calculus.
449
00:37:19,400 --> 00:37:25,940
So calculus can be mentioned, integration
can be mentioned now in the next minute and
450
00:37:25,940 --> 00:37:28,360
not again for weeks.
451
00:37:28,360 --> 00:37:33,440
It's not that I don't like 18.01, but18.06
is better.
452
00:37:33,440 --> 00:37:42,340
Okay. So, -- so what's -- what's the calculus
formula that looks like?
453
00:37:42,350 --> 00:37:50,589
It looks like -- if we were in calculus instead
of summing stuff, we would integrate.
454
00:37:50,589 --> 00:37:56,800
So I would integrate x squared and I would
get one third x
455
00:37:56,800 --> 00:38:07,430
cubed. So if that was like an integral from
one to n, of x squared b x, if the answer
456
00:38:07,430 --> 00:38:13,690
would be one third n cubed -- and it's correct
for the sum also, because that's, like, the
457
00:38:13,690 --> 00:38:14,900
whole point of calculus.
458
00:38:14,900 --> 00:38:19,315
The whole point of calculus is -- oh, I don't
want to tell you the whole -- I mean, you
459
00:38:19,315 --> 00:38:21,280
know the whole point of calculus.
460
00:38:21,280 --> 00:38:27,990
Calculus is like sums except it's continuous.
461
00:38:27,990 --> 00:38:31,950
Okay. And algebra is discreet.
462
00:38:31,950 --> 00:38:32,950
Okay.
463
00:38:32,950 --> 00:38:35,010
So the answer is one third n cubed.
464
00:38:35,010 --> 00:38:40,080
Now I'll just -- let me say one more thing
about operations.
465
00:38:40,080 --> 00:38:41,920
What about the right-hand side?
466
00:38:41,920 --> 00:38:45,250
This was what it cost on the left side.
467
00:38:45,250 --> 00:38:50,380
This is on A.
468
00:38:50,380 --> 00:38:52,180
Because this is A that we're working with.
469
00:38:52,180 --> 00:39:00,660
But what's the cost on the extra column vector
b that we're hanging around here?
470
00:39:00,660 --> 00:39:07,570
So b costs a lot less, obviously, because
it's just one column.
471
00:39:07,570 --> 00:39:14,260
We carry it through elimination and then actually
we do back substitution.
472
00:39:14,260 --> 00:39:16,160
Let me just tell you the answer there.
473
00:39:16,160 --> 00:39:18,420
It's n squared.
474
00:39:18,420 --> 00:39:23,220
So the cost for every right hand side is n
squared.
475
00:39:23,220 --> 00:39:37,000
So let me -- I'll just fit that in here -- for
the cost of b turns out to be n squared.
476
00:39:37,000 --> 00:39:49,580
So you see if we have, as we often have, a
a matrix A and several right-hand sides, then
477
00:39:49,580 --> 00:39:57,670
we pay the price on A, the higher price on
A to get it split up into L and U to do elimination
478
00:39:57,670 --> 00:40:02,810
on A, but then we can process every right-hand
side at low cost.
479
00:40:02,810 --> 00:40:03,859
Okay.
480
00:40:03,860 --> 00:40:16,560
So the -- We really have discussed the most
fundamental algorithm for a system of equations.
481
00:40:16,560 --> 00:40:19,700
Okay.
482
00:40:19,700 --> 00:40:29,500
So, I'm ready to allow row exchanges.
483
00:40:29,500 --> 00:40:34,220
I'm ready to allow -- now what happens to
this whole -- today's lecture if there are
484
00:40:34,220 --> 00:40:38,400
row exchanges?
485
00:40:38,400 --> 00:40:42,240
When would there be row exchanges?
486
00:40:42,240 --> 00:40:47,840
There are row -- we need to do row exchanges
if a zero shows up in the pivot position.
487
00:40:47,840 --> 00:40:55,820
So moving then into the final section of this
chapter, which is about transposes -- well,
488
00:40:55,820 --> 00:41:08,640
we've already seen some transposes, and -- the
title of this section is,
489
00:41:08,640 --> 00:41:13,700
"Transposes
and Permutations."
490
00:41:13,700 --> 00:41:21,160
Okay. So can I say, now, where does a permutation
come in?
491
00:41:21,160 --> 00:41:23,300
Let me talk a little about permutations.
492
00:41:23,300 --> 00:41:34,620
So that'll be up here, permutations.
493
00:41:34,620 --> 00:41:41,420
So these are the matrices that I need to do
row exchanges.
494
00:41:41,420 --> 00:41:44,560
And I may have to do two row exchanges.
495
00:41:44,560 --> 00:41:52,320
Can you invent a matrix where I would have
to do two row exchanges and then would come
496
00:41:52,320 --> 00:41:54,180
out fine?
497
00:41:54,180 --> 00:42:01,460
Yeah let's just, for the heck of it -- so
I'll put it here.
498
00:42:01,460 --> 00:42:04,120
Let me do three by threes.
499
00:42:04,120 --> 00:42:10,480
Actually, why don't I just plain list all
the three by three permutation matrices.
500
00:42:10,480 --> 00:42:13,000
There're a nice little group of them.
501
00:42:13,010 --> 00:42:20,880
What are all the matrices that exchange no
rows at all?
502
00:42:20,880 --> 00:42:26,070
Well, I'll include the identity.
503
00:42:26,070 --> 00:42:29,710
So that's a permutation matrix that doesn't
do anything.
504
00:42:29,710 --> 00:42:38,680
Now what's the permutation matrix that exchanges
-- what is P12? The permutation matrix that
505
00:42:38,680 --> 00:42:48,540
exchanges rows one and two would be -- 0 1
0 -- 1 0 0, right.
506
00:42:48,540 --> 00:42:53,360
I just exchanged those rows of the identity
and I've got it.
507
00:42:53,360 --> 00:42:56,080
Okay. Actually, I'll -- yes.
508
00:42:56,080 --> 00:43:01,840
Let me clutter this up.
509
00:43:01,840 --> 00:43:06,340
Okay. Give me a complete list of all the row
exchange matrices.
510
00:43:06,340 --> 00:43:07,540
So what are they?
511
00:43:07,540 --> 00:43:13,780
They're all the ways I can take the identity
matrix and rearrange its rows.
512
00:43:13,780 --> 00:43:16,620
How many will there be?
513
00:43:16,620 --> 00:43:21,820
How many three by three permutation matrices?
514
00:43:21,820 --> 00:43:24,100
Shall we keep going and get the answer?
515
00:43:24,100 --> 00:43:27,500
So tell me some more.
516
00:43:27,500 --> 00:43:28,580
STUDENT: Zero one –
517
00:43:28,580 --> 00:43:31,000
STRANG: Zero – What one
are you going to do now?
518
00:43:31,000 --> 00:43:32,840
STUDENT: I'm going to switch the –
519
00:43:32,840 --> 00:43:43,520
STRANG: Switch rows one and -- One and
three, okay. One and three, leaving two alone.
520
00:43:43,520 --> 00:43:44,460
Okay.
521
00:43:44,460 --> 00:43:50,780
Now what else? Switch -- what would
be the next easy one -- is switch two and
522
00:43:50,780 --> 00:43:57,240
three, good. So I'll leave one zero zero alone
and I'll switch -- I'll move number three
523
00:43:57,250 --> 00:44:00,329
up and number two down.
524
00:44:00,329 --> 00:44:05,380
Okay. Those are the ones that just exchange
single -- a pair of
525
00:44:05,380 --> 00:44:13,339
rows. This guy, this guy and this guy exchanges
a pair of rows, but now there are more possibilities.
526
00:44:13,340 --> 00:44:16,080
What's left?
527
00:44:16,080 --> 00:44:19,640
So tell -- there is another one here.
528
00:44:19,640 --> 00:44:22,260
What's that?
529
00:44:22,260 --> 00:44:26,440
It's going to move -- it's going
to change all rows, right?
530
00:44:26,440 --> 00:44:28,160
Where shall we put them?
531
00:44:28,160 --> 00:44:29,780
So -- give me a first row.
532
00:44:29,780 --> 00:44:30,800
STUDENT: Zero one zero?
533
00:44:30,800 --> 00:44:32,740
STRANG: Zero one zero.
534
00:44:32,740 --> 00:44:38,000
Okay, now a second row -- say zero zero one
and the third guy
535
00:44:38,000 --> 00:44:41,960
One zero zero.
536
00:44:41,960 --> 00:44:44,420
So that is like a cycle.
537
00:44:44,420 --> 00:44:49,800
That puts row two moves up to row one,
row three moves up to
538
00:44:49,800 --> 00:44:53,420
row two and row one moves
down to row three.
539
00:44:53,420 --> 00:45:00,180
And there's one more, which is -- let's see.
540
00:45:00,180 --> 00:45:03,280
What's left?
541
00:45:03,280 --> 00:45:04,160
I'm lost.
542
00:45:04,160 --> 00:45:05,240
STUDENT: Is it zero zero one?
543
00:45:05,240 --> 00:45:06,780
STRANG: Is it zero zero one? Okay.
544
00:45:06,780 --> 00:45:08,120
STUDENT: One zero zero.
545
00:45:08,120 --> 00:45:11,260
STRANG: One zero zero, okay.
546
00:45:11,260 --> 00:45:16,460
Zero one zero, okay.
547
00:45:16,460 --> 00:45:17,980
Great.
548
00:45:17,980 --> 00:45:21,780
Six. Six of them.
549
00:45:21,780 --> 00:45:35,580
Six P. And they're sort of nice, because what
happens if I write, multiply two of them together?
550
00:45:35,580 --> 00:45:42,100
If I multiply two of these matrices together,
what can you tell me about the answer?
551
00:45:42,100 --> 00:45:44,360
It's on the list.
552
00:45:44,360 --> 00:45:49,220
If I do some row exchanges and then I do some
more row exchanges, then all together I've
553
00:45:49,220 --> 00:45:50,520
done row exchanges.
554
00:45:50,520 --> 00:45:54,520
So if I multiply -- but, I don't know.
555
00:45:54,520 --> 00:46:00,340
And if I invert, then I'm just doing row exchanges
to get back again.
556
00:46:00,349 --> 00:46:01,869
So the inverses are all there.
557
00:46:01,869 --> 00:46:12,790
It's a little family of matrices that -- they've
got their own -- if I multiply, I'm still
558
00:46:12,790 --> 00:46:14,400
inside this group.
559
00:46:14,400 --> 00:46:18,920
If I invert I'm inside this group -- actually,
group is the right name for this subject.
560
00:46:18,920 --> 00:46:24,320
It's a group of six matrices, and what about
the inverses?
561
00:46:24,320 --> 00:46:28,160
What's the inverse of this guy, for example?
562
00:46:28,160 --> 00:46:34,700
What's the inverse -- if I exchange rows one
and two, what's the inverse matrix?
563
00:46:34,700 --> 00:46:36,760
Just tell me fast.
564
00:46:36,760 --> 00:46:46,910
The inverse of that matrix is -- if I exchange
rows one and two, then what I should do to
565
00:46:46,910 --> 00:46:51,150
get back to where I started is the same thing.
566
00:46:51,150 --> 00:46:54,905
So this thing is its own inverse.
567
00:46:54,905 --> 00:46:55,905
That's probably its own inverse.
568
00:46:55,905 --> 00:47:00,380
This is probably not -- actually, I think
these are inverses of each other.
569
00:47:00,380 --> 00:47:06,320
Oh, yeah, actually -- the inverse is the transpose.
570
00:47:06,320 --> 00:47:16,020
There's a curious fact about permutations
matrices, that the inverses are the transposes.
571
00:47:16,020 --> 00:47:21,820
And final moment -- how many are there if
I -- how many four by four permutations?
572
00:47:21,820 --> 00:47:29,880
So let me take four by four -- how many Ps?
573
00:47:29,880 --> 00:47:32,720
Well, okay.
574
00:47:32,720 --> 00:47:34,940
Make a good guess.
575
00:47:34,940 --> 00:47:38,020
Twenty four, right. Twenty four Ps.
576
00:47:38,020 --> 00:47:48,820
Okay. So, we've got these permutation matrices,
and in the next lecture, we'll use them.
577
00:47:48,820 --> 00:47:51,660
So the next lecture,
finishes Chapter 2
578
00:47:51,660 --> 00:47:55,160
and moves to Chapter 3.
579
00:47:55,160 --> 00:47:57,480
Thank you.