WEBVTT
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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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it is now.
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What's the inverse of a product?
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If I multiply two matrices together and I
know their inverses, how do I get the inverse
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of A times B?
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So I know what inverses mean for a single
matrix A and for a matrix B.
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What matrix do I multiply by to get the identity
if I have A B?
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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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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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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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We can just check that A B times that matrix
gives the identity.
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Okay.
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So why -- once again, it's this fact that
I can move parentheses around.
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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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identity.
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So forgive the dumb example in the book.
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Why do you, do the inverse things in reverse
order?
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It's just like -- you take off your shoes,
you take off your socks, then the good way
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to invert that process is socks back on first,
then shoes.
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Sorry, okay.
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I'm sorry that's on the tape.
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And, of course, on the other side we should
really just check -- on the other side I have
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B inverse,
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A inverse. That does multiply A B, and this
time it's these guys that give the identity,
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squeeze down, they give the identity, we're
in shape.
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Okay. So there's the inverse.
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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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So how do I -- if I transpose a matrix, I'm
talking about square, invertible matrices
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right now.
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If I transpose one, what's its inverse?
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Well, the nice formula is -- let's see.
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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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So if I transpose the identity matrix, what
do I have?
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The identity, right?
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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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If I transpose these guys, that product, then
again it turns out that I have to reverse
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the order.
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I can transpose them separately, but when
I multiply, those transposes come in the opposite
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order.
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So it's A inverse transpose times A transpose
giving the identity.
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So that's -- this equation is -- just comes
directly from that
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one. But this equation tells me what I wanted
to know, namely what is the inverse of this
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guy A transpose?
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What's the inverse of that -- if I transpose
a matrix, what'ss the inverse of the result?
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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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So I'll put a big circle around that, because
that's the answer, that's the best answer
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we could hope for.
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That if you want to know the inverse of A
transpose and you know the inverse of A, then
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you just transpose that.
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So in a -- to put it another way, transposing
and inversing you can do in either order for
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a single matrix.
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Okay. So these are like basic facts that we
can now use, all right -- so now I put it
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to use.
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I put it to use by thinking -- we're really
completing, the subject of elimination.
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Actually, -- the thing about elimination is
it's the right way to understand what the
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matrix has got.
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This A equal L U is the most basic factorization
of a matrix.
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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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And please don't.
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We'll be beyond that, but it's the right algebra
to do first.
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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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So my decent form is matrix form.
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I have a matrix A, let's suppose it's a good
matrix, I can do elimination, no row exchanges
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-- So no row exchanges for now.
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Pivots all fine, nothing zero in the pivot
position.
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I get to the very end, which is U.
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So I get from A to U.
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And I want to know what's the connection?
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How is A related to U?
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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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Can I do it for a two by two first?
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Okay. Two by two, elimination.
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Okay, so I'll do it under here.
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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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here and what number do I not want to put
there?
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Four. I don't want a four there, because in
that case, the second pivot would not -- we
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wouldn't have a second pivot.
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The matrix would be singular, general screw-up.
Okay.
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So let me put some other number here like
seven.
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Okay.
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Okay. Now I want to operate on that with my
elementary matrix.
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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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And what's the matrix that did it?
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Quick review, then.
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What's the elimination elementary matrix E21
-- it's one zero, thanks.
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And -- negative four one,
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right. Good.
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Okay. So that -- you see the difference between
this and what I'm shooting for.
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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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So I can do that right away.
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Now here's going to be my A equals L U.
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And you won't have any trouble telling me
what -- so A is still two one eight seven.
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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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Well, first -- so how is L related to this
E guy?
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It's the inverse, because I want to multiply
through by the inverse of this, which will
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put the identity here, and the inverse will
show up there and I'll call it L.
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So what is the inverse of this?
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Remember those elimination matrices are easy
to invert.
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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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Okay. Do you want -- if we did the numbers
right, we must -- this should be correct.
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Okay. And of course it is.
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That says the first row's right, four times
the first row plus the second row is eight
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seven. Good. Okay.
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That's simple, two by two.
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But it already shows the form that we're headed
for.
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It shows -- so what's the L stand for?
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Why the letter L?
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If U stood for upper triangular, then of course
L stands for lower triangular.
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And actually, it has ones on the diagonal,
where this thing has the pivots on the diagonal.
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Oh, sometimes we may want to separate out
the pivots, so can I just mention that sometimes
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we could also write this as -- we could have
this one zero four one -- I'll just show you
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how I would divide out this matrix of pivots
-- two three.
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There's a diagonal matrix.
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And I just -- whatever is left is here.
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Now what's left?
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If I divide this first row by two to pull
out the two, then I have a one and a one half.
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And if I divide the second row by three to
pull out the three, then I have a one.
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So if this is L
U, this is maybe called L D or pivot U.
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And now it's a little more balanced, because
we have ones on the diagonal here and here.
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And the diagonal matrix in the middle.
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So both of those...
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Matlab would produce either one.
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I'll basically stay with L U.
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Okay. Now I have to think about bigger than
two by two.
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But right now, this was just like easy exercise.
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And, to tell the truth, this one was a minus
sign and this one was a plus sign.
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I mean, that's the only difference.
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But, with three by three, there's a more significant
difference.
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Let me show you how that works.
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Let me move up to a three by three, let's
say some matrix A, okay?
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Let's imagine it's three by three.
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I won't write numbers down for now.
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So what's the first elimination step that
I do, the first matrix I multiply it by, what
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letter will I use for
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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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one position. right?
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And then the next step will be to get a zero
in the three one position.
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And the final step will be to get a zero in
the three two
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That's what elimination is, and it produced
U. position.
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And again, no row exchanges.
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I'm taking the nice case, now, the typical
case, too -- when I don't have to do any row
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exchange, all I do is these elimination steps.
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Okay.
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Now, suppose I want that stuff over on the
right-hand side, as I really do.
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That's, like, my point here.
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I can multiply these together to get a matrix
E, but I want it over on the right.
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I want its inverse over there.
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So what's the right expression now?
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If I write A and U, what goes there?
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Okay.
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So I've got the inverse of this, I've got
three matrices in
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a row now.
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And it's their inverses that are going to
show up, because each one is easy to invert.
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Question is, what about the whole bunch?
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How easy is it to invert the whole bunch?
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So, that's what we know how to do.
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We know how to invert, we should take the
separate inverses, but they go in the opposite
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order.
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So what goes here?
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E three two inverse, right, because I'll multiply
from the left by E three two inverse, then
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I'll pop it up next to U.
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And then will come E three one inverse.
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And then this'll be the only guy left standing
and that's gone when I do an E two one inverse.
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So there is L.
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That's L U.
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L is product of inverses.
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Now you still can ask why is this guy preferring
inverses?
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And let me explain why.
00:16:59.570 --> 00:17:05.619
Let me explain why is this product nicer than
this one?
00:17:05.619 --> 00:17:10.380
This product turns out to be better than this
one.
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Let me take a typical case here.
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Let me take a typical case.
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So let me -- I have to do three by three for
you to see the improvement.
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Two by two, it was just one E, no problem.
00:17:27.400 --> 00:17:29.960
But let me go up to this case.
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Suppose my matrices E21 -- suppose E21 has
a minus two in there.
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Suppose that -- and now suppose -- oh, I'll
even suppose E31 is the identity.
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I'm going to make the point with just a couple
of these.
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Okay.
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Now this guy will have -- do something -- now
let's suppose minus five one.
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Okay. There's typical.
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That's a typical case in which we didn't need
an E31. Maybe we already had a zero in that
00:18:12.760 --> 00:18:14.660
three one position.
00:18:14.660 --> 00:18:17.560
Okay.
00:18:17.560 --> 00:18:24.700
Let me see -- is that going to be enough to,
show my point?
00:18:24.700 --> 00:18:29.500
Let me do that multiplication.
00:18:29.500 --> 00:18:32.640
So if I do that multiplication it's like good
practice to
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multiply these matrices.
00:18:35.000 --> 00:18:39.900
Tell me what's above the diagonal when I do
this multiplication?
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
00:18:44.500 --> 00:18:48.280
zeroes above.
00:18:48.280 --> 00:18:50.140
Because -- what does that say?
00:18:50.140 --> 00:18:54.540
That says that I'm subtracting rows from lower
rows.
00:18:54.549 --> 00:19:00.970
So nothing is moving upwards as it did last
time in Gauss-Jordan. Okay.
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
00:19:10.280 --> 00:19:11.240
that number?
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This is the number that I'm really have in
mind.
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That number is ten.
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And this one is -- what goes here?
00:19:25.620 --> 00:19:28.960
Row three against column two, it looks like
the minus five.
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What – it's that ten.
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How did that ten get in there?
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I don't like that ten.
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I mean -- of course, I don't want to erase
it, because it's right.
00:19:42.220 --> 00:19:45.840
But I don't want it there.
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,
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and then I subtracted five of that new row
two from row three.
00:19:58.760 --> 00:20:04.920
So doing it in that order, how did row one
effect row three?
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
00:20:10.480 --> 00:20:11.590
from row three.
00:20:11.590 --> 00:20:17.800
So altogether ten of row one got thrown into
row three.
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Now my point is in the reverse direction -- so
now I can do it -- below it I'll do the inverses.
00:20:27.540 --> 00:20:28.100
Okay.
00:20:28.100 --> 00:20:29.820
And, of course, opposite order.
00:20:29.820 --> 00:20:31.660
Reverse order.
00:20:31.660 --> 00:20:35.960
Reverse order.
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.
00:20:44.020 --> 00:20:47.790
Left of A.
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
00:20:54.340 --> 00:20:57.530
means I put this inverse first.
00:20:57.530 --> 00:20:58.890
And what is its inverse?
00:20:58.890 --> 00:21:04.660
What's the inverse of E21? Same thing with
a plus sign, right?
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
00:21:12.730 --> 00:21:16.620
two, so no problem.
00:21:16.620 --> 00:21:21.400
And now, in reverse order, I want to invert
that.
00:21:21.400 --> 00:21:22.840
Just right?
00:21:22.840 --> 00:21:25.540
I'm doing just this, this.
00:21:25.540 --> 00:21:36.320
So now the inverse is again the same thing,
but add in the five.
00:21:36.320 --> 00:21:45.460
And now I'll do that multiplication and I'll
get a happy result.
00:21:45.460 --> 00:21:47.800
I hope.
00:21:47.800 --> 00:21:50.220
Did I do it right so far?
00:21:50.220 --> 00:21:50.960
Yes, okay.
00:21:50.960 --> 00:21:51.640
Let me do the multiplication.
00:21:51.650 --> 00:21:53.350
I believe this comes out.
00:21:53.350 --> 00:21:56.660
So row one of the answer is one zero zero.
00:21:56.660 --> 00:21:59.300
Oh, I know that all this is going to be left,
00:21:59.300 --> 00:22:04.080
right? Then I have two one zero.
00:22:04.090 --> 00:22:07.559
So I get two one zero there, right?
00:22:07.560 --> 00:22:10.060
And what's the third row?
00:22:10.060 --> 00:22:15.160
What's the third row in this product?
00:22:15.160 --> 00:22:18.620
Just read it out to me, the third row?
00:22:18.640 --> 00:22:22.980
0 5 1
00:22:22.980 --> 00:22:27.240
Because one way to say is –
this is saying take one of the
00:22:27.240 --> 00:22:29.640
last row and there it is.
00:22:29.640 --> 00:22:32.580
And this is my matrix L.
00:22:32.590 --> 00:22:38.450
And it's the one that goes on the left of
U.
00:22:38.450 --> 00:22:45.190
It goes into -- what do I mean here?
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.
00:22:52.040 --> 00:22:58.600
E A is U, whereas A is L U.
00:22:58.600 --> 00:23:01.560
Okay.
00:23:01.560 --> 00:23:06.280
Let me make the point now in words.
00:23:06.300 --> 00:23:11.090
The order that the matrices come for L is
the right order.
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
00:23:21.800 --> 00:23:26.919
right order, the multipliers just sit in the
matrix L.
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.
00:23:35.530 --> 00:23:41.370
I just keep a record of what those multipliers
were, and that gives me L.
00:23:41.370 --> 00:23:52.610
So I'll draw the -- let me say it.
00:23:52.610 --> 00:23:56.320
So this is the A=L U.
00:23:56.320 --> 00:24:14.010
So if no row exchanges, the multipliers that
those numbers that we multiplied rows by and
00:24:14.010 --> 00:24:27.550
subtracted, when we did an elimination step
-- the multipliers go directly into L.
00:24:27.550 --> 00:24:29.950
Okay.
00:24:29.950 --> 00:24:42.770
So L is -- this is the way, to look at elimination.
00:24:42.770 --> 00:24:51.510
You go through the elimination steps, and
actually if you do it right, you can throw
00:24:51.510 --> 00:24:57.120
away A as you create L U.
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,
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
00:25:16.540 --> 00:25:22.980
that you used -- which you have to save,
and then you can forget A.
00:25:22.980 --> 00:25:25.919
So because it's all there in L and U.
00:25:25.919 --> 00:25:41.580
So that's -- this moment is maybe the new
insight in elimination that comes from matrix
00:25:41.580 --> 00:25:44.070
-- doing it in matrix form.
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.
00:25:52.350 --> 00:25:56.350
The matrix E is not a particularly attractive
one.
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,
00:26:02.130 --> 00:26:06.210
there the L comes out just right. Okay.
00:26:06.210 --> 00:26:09.580
Now -- oh gosh, so today's a sort of,
00:26:09.580 --> 00:26:14.160
like, practical day.
00:26:14.160 --> 00:26:20.670
Can we think together how expensive is elimination?
00:26:20.670 --> 00:26:24.570
How many operations do we do?
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
00:26:32.540 --> 00:26:34.980
it came. Here it comes.
00:26:34.980 --> 00:26:55.780
How many operations on an n by n matrix A.
00:26:55.780 --> 00:26:58.020
I mean, it's a very practical question.
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?
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?
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.
00:27:25.059 --> 00:27:28.700
I mean, we've put in more information.
00:27:28.700 --> 00:27:33.880
We make the whole thing is more accurate for
the bigger matrix.
00:27:33.880 --> 00:27:39.000
But it's more expensive, too, and the question
is how much more expensive?
00:27:39.000 --> 00:27:41.679
If I have matrices of order a hundred.
00:27:41.679 --> 00:27:43.150
Let's say a hundred by a hundred.
00:27:43.150 --> 00:27:47.660
Let me take n to be a hundred.
00:27:47.660 --> 00:27:50.840
Say n equal a hundred.
00:27:50.840 --> 00:27:54.860
How many steps are we doing?
00:27:54.860 --> 00:28:02.770
How many operations are we actually doing
that we -- And let's suppose there aren't
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
00:28:07.800 --> 00:28:09.870
don't have to do those operations, and,
00:28:09.870 --> 00:28:13.500
it'll be much faster.
00:28:13.500 --> 00:28:21.940
But -- so just think for a moment about the
first step.
00:28:21.940 --> 00:28:27.179
So here's our matrix A, hundred by a hundred.
00:28:27.180 --> 00:28:34.880
And the first step will be -- that column,
is got zeroes down
00:28:34.880 --> 00:28:41.080
here. So it's down to 99 by 99, right?
00:28:41.080 --> 00:28:45.920
That's really like the first stage of elimination,
00:28:45.920 --> 00:28:48.480
to get from this hundred
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
00:28:55.600 --> 00:28:59.440
first row's okay the first column is okay.
00:28:59.440 --> 00:29:05.120
So, eventually -- how many steps did that
take?
00:29:05.120 --> 00:29:07.400
You see, I'm trying to get an idea.
00:29:07.409 --> 00:29:11.860
Is the answer proportional to n?
00:29:11.860 --> 00:29:17.260
Is the total number of steps in elimination,
the total number, is it proportional to n
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
00:29:22.520 --> 00:29:24.140
long?
00:29:24.140 --> 00:29:27.340
Does it square, so it would take me four times
as long?
00:29:27.350 --> 00:29:30.600
Does it cube so it would take me eight times
as long?
00:29:30.600 --> 00:29:37.120
Or is it n factorial, so it would take me
a hundred times as long?
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
00:29:41.560 --> 00:29:44.280
cost, here.
00:29:44.280 --> 00:29:48.720
So these are the questions that I'm -- let
me ask those questions again.
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
00:29:54.900 --> 00:29:56.420
power of n?
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
00:30:04.580 --> 00:30:07.980
and then by a hundred and two -- which is
it?
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
00:30:14.320 --> 00:30:16.100
had to do.
00:30:16.100 --> 00:30:17.100
Okay.
00:30:17.100 --> 00:30:22.680
So what was the cost here?
00:30:22.680 --> 00:30:23.300
Well, let's see.
00:30:23.309 --> 00:30:25.700
What do I mean by an operation?
00:30:25.700 --> 00:30:32.549
I guess I mean, well an addition or -- yeah.
00:30:32.549 --> 00:30:33.549
No big deal.
00:30:33.549 --> 00:30:38.980
I guess I mean an addition or a subtraction
or a multiplication or a division.
00:30:38.980 --> 00:30:41.540
Okay.
00:30:41.540 --> 00:30:48.980
And actually, what operation I doing all the
time?
00:30:48.980 --> 00:30:58.870
When I multiply row one by multiplier L and
I subtract from row six.
00:30:58.870 --> 00:31:01.060
What's happening there individually?
00:31:01.060 --> 00:31:03.460
What's going on?
00:31:03.460 --> 00:31:08.490
If I multiply -- I do a multiplication by
L and then a subtraction.
00:31:08.490 --> 00:31:15.610
So I guess operation -- Can I count that for
the moment as, like, one operation?
00:31:15.610 --> 00:31:18.520
Or you may want to count them separately.
00:31:18.520 --> 00:31:26.740
The typical operation is multiply plus a subtract.
00:31:26.740 --> 00:31:31.420
So if I count those together, my answer's
going to come out
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
00:31:36.540 --> 00:31:39.000
number of multiplies, certain number of subtracts.
00:31:39.000 --> 00:31:40.580
That's really want to do.
00:31:40.580 --> 00:31:43.360
Okay. How many have I got here?
00:31:43.360 --> 00:31:49.500
So, I think -- let's see.
00:31:49.500 --> 00:31:56.293
It's about -- well, how many, roughly?
00:31:56.293 --> 00:31:59.352
How many operations to get from here to here?
00:31:59.360 --> 00:32:05.800
Well, maybe one way to look at it is all these
numbers had to get changed.
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.
00:32:11.900 --> 00:32:25.880
So this step -- well, I guess maybe -- shall
I say it cost about a hundred squared.
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,
00:32:32.110 --> 00:32:35.549
because -- because that's how many numbers
are here.
00:32:35.549 --> 00:32:42.279
A hundred squared numbers is the total count
of the entry, and all but this insignificant
00:32:42.279 --> 00:32:44.299
first row got changed.
00:32:44.300 --> 00:32:47.660
So I would say about a hundred squared.
00:32:47.660 --> 00:32:48.620
Okay.
00:32:48.620 --> 00:32:54.150
Now, what about the next step?
00:32:54.150 --> 00:32:56.960
So now the first row is fine.
00:32:56.960 --> 00:33:00.020
The second row is fine.
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?
00:33:06.700 --> 00:33:08.620
And then you're with me.
00:33:08.620 --> 00:33:10.380
Roughly, what's the cost?
00:33:10.390 --> 00:33:17.330
If this first step cost a hundred squared,
about, operations then this one, which is
00:33:17.330 --> 00:33:27.490
really working on this guy to produce this,
costs about what?
00:33:27.490 --> 00:33:31.250
How many operations to fix?
00:33:31.250 --> 00:33:36.610
About ninety-nine squared, or ninety-nine
times ninety-eight. But less, right?
00:33:36.610 --> 00:33:39.590
Less, because our problem's getting smaller.
00:33:39.590 --> 00:33:42.280
About ninety-nine squared.
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
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.
00:33:53.750 --> 00:33:55.890
The big numbers are here.
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?
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
00:34:13.589 --> 00:34:23.040
down to three squared and two squared and
even one squared.
00:34:23.040 --> 00:34:27.929
No way I should have written that -- squeezed
that in.
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
00:34:37.570 --> 00:34:41.980
to one squared.
00:34:41.980 --> 00:34:44.899
That's a pretty decent count.
00:34:44.899 --> 00:34:56.040
Admittedly, we didn't catch every single tiny
operation, but we got the right leading term
00:34:56.040 --> 00:34:57.040
here.
00:34:57.040 --> 00:35:01.180
And what do those add up to?
00:35:01.180 --> 00:35:10.030
Okay, so now we're coming to the punch of
this, question, this operation count.
00:35:10.030 --> 00:35:19.820
So the operations on the left side, on the
matrix A to finally get to U.
00:35:19.820 --> 00:35:28.540
And anybody -- so which of these quantities
is the right ballpark for that count?
00:35:28.540 --> 00:35:34.080
If I add a hundred squared to ninety nine
squared to ninety eight squared -- ninety
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?
00:35:44.020 --> 00:35:48.100
It's just one of these -- let's identify it
first.
00:35:48.100 --> 00:35:49.020
Is it n?
00:35:49.020 --> 00:35:52.080
Certainly not.
00:35:52.080 --> 00:35:55.700
Is it n factorial?
00:35:55.700 --> 00:35:56.460
No.
00:35:56.460 --> 00:36:02.020
If it was n factorial, we would -- with determinants,
it is n factorial.
00:36:02.020 --> 00:36:12.400
I'll put in a bad mark against determinants,
because that -- okay, so what is it?
00:36:12.400 --> 00:36:18.580
It's n -- well, this is the answer.
00:36:18.580 --> 00:36:21.240
It's this order -- n cubed.
00:36:21.240 --> 00:36:26.380
It's like I have n terms, right?
00:36:26.380 --> 00:36:28.680
I've got n terms in this sum.
00:36:28.690 --> 00:36:31.140
And the biggest one is n squared.
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
00:36:40.750 --> 00:36:45.270
-- it's about one third of n cubed.
00:36:45.270 --> 00:36:53.690
That's the magic operation count.
00:36:53.690 --> 00:37:02.339
Somehow that one third takes account of the
fact that the numbers are getting smaller.
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
00:37:06.900 --> 00:37:08.240
be exactly n cubed.
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.
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.
00:37:19.400 --> 00:37:25.940
So calculus can be mentioned, integration
can be mentioned now in the next minute and
00:37:25.940 --> 00:37:28.360
not again for weeks.
00:37:28.360 --> 00:37:33.440
It's not that I don't like 18.01, but18.06
is better.
00:37:33.440 --> 00:37:42.340
Okay. So, -- so what's -- what's the calculus
formula that looks like?
00:37:42.350 --> 00:37:50.589
It looks like -- if we were in calculus instead
of summing stuff, we would integrate.
00:37:50.589 --> 00:37:56.800
So I would integrate x squared and I would
get one third x
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
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
00:38:13.690 --> 00:38:14.900
whole point of calculus.
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
00:38:19.315 --> 00:38:21.280
know the whole point of calculus.
00:38:21.280 --> 00:38:27.990
Calculus is like sums except it's continuous.
00:38:27.990 --> 00:38:31.950
Okay. And algebra is discreet.
00:38:31.950 --> 00:38:32.950
Okay.
00:38:32.950 --> 00:38:35.010
So the answer is one third n cubed.
00:38:35.010 --> 00:38:40.080
Now I'll just -- let me say one more thing
about operations.
00:38:40.080 --> 00:38:41.920
What about the right-hand side?
00:38:41.920 --> 00:38:45.250
This was what it cost on the left side.
00:38:45.250 --> 00:38:50.380
This is on A.
00:38:50.380 --> 00:38:52.180
Because this is A that we're working with.
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?
00:39:00.660 --> 00:39:07.570
So b costs a lot less, obviously, because
it's just one column.
00:39:07.570 --> 00:39:14.260
We carry it through elimination and then actually
we do back substitution.
00:39:14.260 --> 00:39:16.160
Let me just tell you the answer there.
00:39:16.160 --> 00:39:18.420
It's n squared.
00:39:18.420 --> 00:39:23.220
So the cost for every right hand side is n
squared.
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.
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
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
00:39:57.670 --> 00:40:02.810
on A, but then we can process every right-hand
side at low cost.
00:40:02.810 --> 00:40:03.859
Okay.
00:40:03.860 --> 00:40:16.560
So the -- We really have discussed the most
fundamental algorithm for a system of equations.
00:40:16.560 --> 00:40:19.700
Okay.
00:40:19.700 --> 00:40:29.500
So, I'm ready to allow row exchanges.
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
00:40:34.220 --> 00:40:38.400
row exchanges?
00:40:38.400 --> 00:40:42.240
When would there be row exchanges?
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.
00:40:47.840 --> 00:40:55.820
So moving then into the final section of this
chapter, which is about transposes -- well,
00:40:55.820 --> 00:41:08.640
we've already seen some transposes, and -- the
title of this section is,
00:41:08.640 --> 00:41:13.700
"Transposes
and Permutations."
00:41:13.700 --> 00:41:21.160
Okay. So can I say, now, where does a permutation
come in?
00:41:21.160 --> 00:41:23.300
Let me talk a little about permutations.
00:41:23.300 --> 00:41:34.620
So that'll be up here, permutations.
00:41:34.620 --> 00:41:41.420
So these are the matrices that I need to do
row exchanges.
00:41:41.420 --> 00:41:44.560
And I may have to do two row exchanges.
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
00:41:52.320 --> 00:41:54.180
out fine?
00:41:54.180 --> 00:42:01.460
Yeah let's just, for the heck of it -- so
I'll put it here.
00:42:01.460 --> 00:42:04.120
Let me do three by threes.
00:42:04.120 --> 00:42:10.480
Actually, why don't I just plain list all
the three by three permutation matrices.
00:42:10.480 --> 00:42:13.000
There're a nice little group of them.
00:42:13.010 --> 00:42:20.880
What are all the matrices that exchange no
rows at all?
00:42:20.880 --> 00:42:26.070
Well, I'll include the identity.
00:42:26.070 --> 00:42:29.710
So that's a permutation matrix that doesn't
do anything.
00:42:29.710 --> 00:42:38.680
Now what's the permutation matrix that exchanges
-- what is P12? The permutation matrix that
00:42:38.680 --> 00:42:48.540
exchanges rows one and two would be -- 0 1
0 -- 1 0 0, right.
00:42:48.540 --> 00:42:53.360
I just exchanged those rows of the identity
and I've got it.
00:42:53.360 --> 00:42:56.080
Okay. Actually, I'll -- yes.
00:42:56.080 --> 00:43:01.840
Let me clutter this up.
00:43:01.840 --> 00:43:06.340
Okay. Give me a complete list of all the row
exchange matrices.
00:43:06.340 --> 00:43:07.540
So what are they?
00:43:07.540 --> 00:43:13.780
They're all the ways I can take the identity
matrix and rearrange its rows.
00:43:13.780 --> 00:43:16.620
How many will there be?
00:43:16.620 --> 00:43:21.820
How many three by three permutation matrices?
00:43:21.820 --> 00:43:24.100
Shall we keep going and get the answer?
00:43:24.100 --> 00:43:27.500
So tell me some more.
00:43:27.500 --> 00:43:28.580
STUDENT: Zero one –
00:43:28.580 --> 00:43:31.000
STRANG: Zero – What one
are you going to do now?
00:43:31.000 --> 00:43:32.840
STUDENT: I'm going to switch the –
00:43:32.840 --> 00:43:43.520
STRANG: Switch rows one and -- One and
three, okay. One and three, leaving two alone.
00:43:43.520 --> 00:43:44.460
Okay.
00:43:44.460 --> 00:43:50.780
Now what else? Switch -- what would
be the next easy one -- is switch two and
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
00:43:57.250 --> 00:44:00.329
up and number two down.
00:44:00.329 --> 00:44:05.380
Okay. Those are the ones that just exchange
single -- a pair of
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.
00:44:13.340 --> 00:44:16.080
What's left?
00:44:16.080 --> 00:44:19.640
So tell -- there is another one here.
00:44:19.640 --> 00:44:22.260
What's that?
00:44:22.260 --> 00:44:26.440
It's going to move -- it's going
to change all rows, right?
00:44:26.440 --> 00:44:28.160
Where shall we put them?
00:44:28.160 --> 00:44:29.780
So -- give me a first row.
00:44:29.780 --> 00:44:30.800
STUDENT: Zero one zero?
00:44:30.800 --> 00:44:32.740
STRANG: Zero one zero.
00:44:32.740 --> 00:44:38.000
Okay, now a second row -- say zero zero one
and the third guy
00:44:38.000 --> 00:44:41.960
One zero zero.
00:44:41.960 --> 00:44:44.420
So that is like a cycle.
00:44:44.420 --> 00:44:49.800
That puts row two moves up to row one,
row three moves up to
00:44:49.800 --> 00:44:53.420
row two and row one moves
down to row three.
00:44:53.420 --> 00:45:00.180
And there's one more, which is -- let's see.
00:45:00.180 --> 00:45:03.280
What's left?
00:45:03.280 --> 00:45:04.160
I'm lost.
00:45:04.160 --> 00:45:05.240
STUDENT: Is it zero zero one?
00:45:05.240 --> 00:45:06.780
STRANG: Is it zero zero one? Okay.
00:45:06.780 --> 00:45:08.120
STUDENT: One zero zero.
00:45:08.120 --> 00:45:11.260
STRANG: One zero zero, okay.
00:45:11.260 --> 00:45:16.460
Zero one zero, okay.
00:45:16.460 --> 00:45:17.980
Great.
00:45:17.980 --> 00:45:21.780
Six. Six of them.
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?
00:45:35.580 --> 00:45:42.100
If I multiply two of these matrices together,
what can you tell me about the answer?
00:45:42.100 --> 00:45:44.360
It's on the list.
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
00:45:49.220 --> 00:45:50.520
done row exchanges.
00:45:50.520 --> 00:45:54.520
So if I multiply -- but, I don't know.
00:45:54.520 --> 00:46:00.340
And if I invert, then I'm just doing row exchanges
to get back again.
00:46:00.349 --> 00:46:01.869
So the inverses are all there.
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
00:46:12.790 --> 00:46:14.400
inside this group.
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.
00:46:18.920 --> 00:46:24.320
It's a group of six matrices, and what about
the inverses?
00:46:24.320 --> 00:46:28.160
What's the inverse of this guy, for example?
00:46:28.160 --> 00:46:34.700
What's the inverse -- if I exchange rows one
and two, what's the inverse matrix?
00:46:34.700 --> 00:46:36.760
Just tell me fast.
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
00:46:46.910 --> 00:46:51.150
get back to where I started is the same thing.
00:46:51.150 --> 00:46:54.905
So this thing is its own inverse.
00:46:54.905 --> 00:46:55.905
That's probably its own inverse.
00:46:55.905 --> 00:47:00.380
This is probably not -- actually, I think
these are inverses of each other.
00:47:00.380 --> 00:47:06.320
Oh, yeah, actually -- the inverse is the transpose.
00:47:06.320 --> 00:47:16.020
There's a curious fact about permutations
matrices, that the inverses are the transposes.
00:47:16.020 --> 00:47:21.820
And final moment -- how many are there if
I -- how many four by four permutations?
00:47:21.820 --> 00:47:29.880
So let me take four by four -- how many Ps?
00:47:29.880 --> 00:47:32.720
Well, okay.
00:47:32.720 --> 00:47:34.940
Make a good guess.
00:47:34.940 --> 00:47:38.020
Twenty four, right. Twenty four Ps.
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.
00:47:48.820 --> 00:47:51.660
So the next lecture,
finishes Chapter 2
00:47:51.660 --> 00:47:55.160
and moves to Chapter 3.
00:47:55.160 --> 00:47:57.480
Thank you.