Name: Lec 4 因式化成A=LU。 (Lec 4 Factorization into A = LU)
Uploaded: 2021-01-14T07:52:06.000Z
Duration: 48 min 5 s
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Okay, this is linear algebra, lecture four.

And, the first thing I have to do is something
that was on the list for last time, but here

If I multiply two matrices together and I
know their inverses, how do I get the inverse

So I know what inverses mean for a single
matrix A and for a matrix B.

What matrix do I multiply by to get the identity
if I have A B?

Then I'm going to use that to -- I will have
a product of matrices and the product that

we'll meet will be these elimination matrices
and the net result of today's lectures is

the big formula for elimination, so the net
result of today's lecture is this great way

We know that we get from A to U by elimination.

We know the steps -- but now we get the right
way to look at it, A equals L U.

Can I take the easy part, the first step first?

So, suppose A is invertible -- and of course
it's going to be a big question, when is the

But let's say A is invertible and B is invertible,
then what matrix gives me the inverse of A B?

Yes. I multiply the two matrices A inverse
and B inverse, but what order do I multiply?

So the right thing to put here is B inverse
A inverse.

We can just check that A B times that matrix
gives the identity.

So why -- once again, it's this fact that
I can move parentheses around.

I can just erase them all and do the multiplications
any way I want to.

So what's the right multiplication to do first?

Then A times the identity is the identity
and then finally A times A inverse gives the

Why do you, do the inverse things in reverse
order?

It's just like -- you take off your shoes,
you take off your socks, then the good way

to invert that process is socks back on first,
then shoes.

And, of course, on the other side we should
really just check -- on the other side I have

A inverse. That does multiply A B, and this
time it's these guys that give the identity,

squeeze down, they give the identity, we're
in shape.

Good. While we're at it, let me do a transpose,
because the next lecture has got a lot to

So how do I -- if I transpose a matrix, I'm
talking about square, invertible matrices

Let me start from A, A inverse equal the identity.

That will bring a transpose into the picture.

So if I transpose the identity matrix, what
do I have?

If I exchange rows and columns, the identity
is a symmetric matrix.

If I transpose these guys, that product, then
again it turns out that I have to reverse

I can transpose them separately, but when
I multiply, those transposes come in the opposite

So it's A inverse transpose times A transpose
giving the identity.

So that's -- this equation is -- just comes
directly from that

one. But this equation tells me what I wanted
to know, namely what is the inverse of this

What's the inverse of that -- if I transpose
a matrix, what'ss the inverse of the result?

And this equation tells me that here it is.

So I'll put a big circle around that, because
that's the answer, that's the best answer

That if you want to know the inverse of A
transpose and you know the inverse of A, then

So in a -- to put it another way, transposing
and inversing you can do in either order for

Okay. So these are like basic facts that we
can now use, all right -- so now I put it

I put it to use by thinking -- we're really
completing, the subject of elimination.

Actually, -- the thing about elimination is
it's the right way to understand what the

This A equal L U is the most basic factorization
of a matrix.

I always worry that you will think this course
is all elimination.

We'll be beyond that, but it's the right algebra
to do first.

Okay. So, now I'm coming near the end of it,
but I want to get it in a decent form.

I have a matrix A, let's suppose it's a good
matrix, I can do elimination, no row exchanges

Pivots all fine, nothing zero in the pivot
position.

And I want to know what's the connection?

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Lec 4 因式化成A=LU。 (Lec 4 Factorization into A = LU)

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