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I'll get started, then, with transposes
and permutations.

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So let's let A by an M by N matrix.
So, this is a matrix

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with M rows, and N columns.
And then I'm going to.

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Introduce an operation called the
transpose.

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So the transpose of A is just going to be

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denoted by A with a little T in the
subscript.

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So, you know, just like A squared this is
A transpose.

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And that just means so the transpose
operation just puts the columns

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So the columns of A transposed are the
rows of A, so the

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first row of A becomes the first column of
A transposed, the second

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row of A becomes the second column of A
transposed and so on.

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And if you think about it in terms of the
the

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elements so a matrix A has elements that
are little a,

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sub i j.

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In the transpose those would just be a sub
j i so I'm just swapping the indeces.

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And so if I do that if the dimension of my
matrix a is m by n then if

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I swap the row and the columns, the
dimension of a transpose is

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going to be n by m.
So here I had n columns.

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They become the n rows of a transpose.

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And just, I mean it's a, it's a pretty
obvious thing to do, if I have

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a as just one, two, three, four, five, six
so the first row is one, two, three.

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And the first column of A transpose is
one, two, three.

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And when A is not square, not that the
dimension changes too.

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So it goes from being a two by three
matrix.

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So A has two rows and three columns.

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A transpose then has three rows and two
columns.

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And so some properties of the transpose,

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the first one, hope everybody agrees with
this.

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So if I take A transpose and transpose it
again, I just get A back.

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If I have a

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transpose

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of a sum,

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so I have

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A plus B

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transpose,

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thats

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just going

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to be equal to

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[INAUDIBLE]

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it's not

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letting me

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highlight

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there,

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has to

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be the

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same as

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whatever

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is in the

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3 1 position

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and in

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the,

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the 4

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is here

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have to

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be same,

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so that

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would

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be 3 2

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position

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has to

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be the

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same as

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the 2 3

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position So

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here I have a and a transposed being the
same thing, so this is a symmetric matrix.

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And then an even more special case of
symmetric matrices is a diagonal matrix.

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And so a matrix is diagonal if.
Only the diagonal elements are non zero.

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So, any element aij where j is not equal
to i has to be equal to zero,

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and so a diagonal element just because
these are going to be.

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So, this is D11, D22,

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D33 if I switch the order of those
subscripts.

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They stay the same, so diagonal matrix is
automatically symmetric.

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And again, if, the important thing here,
not that

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it's symmetric, that should be pretty
obvious, but that

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when I have a diagonal matrix, if I take

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the transpose of it, I just get the matrix
back.

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OK.

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And so now I want to explain a

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little bit where diagonal matrices are
going to come from.

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Sorry, where symmetric matrices are
going to come from.

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So let's let R be any m by n matrix.

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So here I'm using R just because I want

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it to be a rectangular matrix, so, not
necessarily square.

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So it has m rows and n columns.

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Then the matrix A that I define as the
product of

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R transposed and R is going to be a
symmetric matrix.

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And so to show that all we have to do is

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take the transpose of A and show that
that's equal to A.

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So the way I'm going to do that I'll take
A is equal to R transpose R.

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So then A transpose is just going to

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be the transpose of the product R
transpose R.

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And now I want to use the rules that I
gave a

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couple of slides ago for how I can deal
with transposes of products.

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So basically says I take the transpose of
the

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pieces and I get them back in the opposite
order.

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So this R here becomes this R transposed
here and this R transposed here, well I

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00:08:00,132 --> 00:08:03,614
take the transpose of that So it's R

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00:08:03,614 --> 00:08:06,420
transposed, transposed so it's going to
become R again.

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00:08:07,940 --> 00:08:12,812
So I have R transposed R and that's what I
initially was calling A,

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so I can start out with any rectangular
matrix and if I make this matrix

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R transposed R, then that product A is
going to be symmetric matrix.

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And you can do something similar for A
equals RR transpose.

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Let's see if I, okay, so I go through

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this example again, so it's basically the
same idea.

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I've defined A to be RR transpose.

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And I'm going to take the transpose of
that and if I can show

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that to equal to A, then A transpose
equals to A, so A is symentric.

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And so the same thing happens, it's just
going to swap the two terms.

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So I have R transpose, transposed, so
that's where this first term comes from.

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The first piece of this product.

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And then here I have an R and then I am
going to transpose it, so I end up with R

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transpose, and then the transpose of a
transpose is just I get the matrix back.

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So, that will be R, R transpose which

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is A, so I get RR transpose is also
symmetric.

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And so what we're going to find as we
start looking at matrix

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factorizations is a lot of times were
going to start out with a

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matrix r that's rectangular, And we're
going to do some calculation where we're

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going to end up the product r transpose r,
or r, r transpose.

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Or maybe both of them.

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And so symmetric matrices are going to
show up

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sort of more often than you would expect
just randomly.

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And there are a lot of nice properties of
symmetric

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matrices that I'll get to later in this
set of lectures.

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And the second type of matrix I want to
talk about is called a permutation matrix.

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Again, a permutation matrix is going to be
square.

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So it's a n by n permutation matrix, P,
has the rows

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00:10:09,564 --> 00:10:14,160
of, so this is I, the identity matrix but
in any order.

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So the identity matrix, it has the, you
know, the first

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row, the row that starts with a one, goes
on top.

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The row that has a one in the second
position goes in the second place.

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And if I shuffle those around in any way,
the matrix that

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I end up with is going to be a permutation
matrix.

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And so for a 3 by 3 matrices, there's 6
possible permutation matrices, so.

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You know, one of the things I could get
when I shuffle

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the rows around is, I could just get the
identity matrix back.

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00:10:45,850 --> 00:10:47,635
So we're actually going to count that as a

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permutation matrix that says just put
everything where it is.

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And then these other ones, so the one's
I've labeled 21, 3,1 and 3,2.

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What these

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matrices are going to do when I multiply,
they're just going to swap rows.

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00:11:05,320 --> 00:11:07,980
So where this would be important, you
know,

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when I was talking about pivets when we
doing

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00:11:10,850 --> 00:11:13,720
elimination it's possible that you could
end up

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00:11:13,720 --> 00:11:16,990
with a, you could end up with an equation.

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00:11:16,990 --> 00:11:21,100
There's no particular reason why the
equations have to be in any order.

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00:11:21,100 --> 00:11:24,532
And so if you had an equation that started
out, so you

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had 0 times x plus something times y plus
something times

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00:11:28,276 --> 00:11:33,850
z equals something, and suppose I decided
to put that equation first.

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Then I wouldn't have a pivot in the first
position.

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00:11:36,420 --> 00:11:40,910
So I probably wouldn't want that to be my
first equation in my system.

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00:11:40,910 --> 00:11:44,220
And so this is sort of the matrix way of
solving that problem.

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00:11:44,220 --> 00:11:46,192
If I needed to get that row out of the

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00:11:46,192 --> 00:11:49,330
first row, I could multiply by one of
these matrices.

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00:11:49,330 --> 00:11:49,570
So for

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00:11:49,570 --> 00:11:51,520
instance, I could swap it with the second
row.

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And the other interesting that some of
these have is, if I multiply,

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00:11:58,508 --> 00:12:04,790
P21 by P21, so this is the matrix that
swaps the first two letters.

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00:12:04,790 --> 00:12:08,420
If I do that again, I would get my
original matrix back.

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00:12:08,420 --> 00:12:11,760
So that means that P21 is it's own
inverse.

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So it swaps row one and two but if I do
that again it puts,

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00:12:14,842 --> 00:12:17,589
you know, what was originally row one back
in row

162
00:12:17,589 --> 00:12:20,620
one, and what was originally row two back
in row two.

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00:12:26,000 --> 00:12:30,260
The other interesting property that this
has is P inverse is the same as P

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00:12:30,260 --> 00:12:36,289
transpose.
So for P21, since this is a symmetric

165
00:12:36,289 --> 00:12:43,710
matrix, and it's its own inverse, you have
this strange...

166
00:12:43,710 --> 00:12:45,150
I guess I would call it duality, but

167
00:12:45,150 --> 00:12:48,020
there's actually three things, so maybe
it's triality.

168
00:12:48,020 --> 00:12:51,035
You have the matrix, its transpose, and
its inverse

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00:12:51,035 --> 00:12:52,620
all being the same matrix.

170
00:12:53,746 --> 00:12:57,538
for the more complicated ones, so if you
want to swap

171
00:12:57,538 --> 00:13:01,330
you know, more like pairs of rows then you
still have

172
00:13:01,330 --> 00:13:05,201
this property that pre, p inverse is p
transpose, But you

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00:13:05,201 --> 00:13:10,569
don't have the property that that these
are their own inverses anymore.

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00:13:14,050 --> 00:13:17,166
As it is the example of what this is going
to do, so

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00:13:17,166 --> 00:13:22,240
the permutation matrix P32 that just swaps
the third row and the second row.

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00:13:23,390 --> 00:13:27,789
So if I have a vector 1 2 3, so it's the
column vector, so the

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00:13:27,789 --> 00:13:33,310
first row is just 1 2 3 Then it's just
going to give me 1, 3, 2.

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00:13:33,310 --> 00:13:34,673
So it took what ever was in the third

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00:13:34,673 --> 00:13:36,630
position and will put it in the second
position.

180
00:13:36,630 --> 00:13:39,220
Whatever was in the second position put it
in the third position.

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00:13:42,440 --> 00:13:46,066
And then if I multiply the output of this,
so 1, 3, 2 by

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00:13:46,066 --> 00:13:50,840
the same permutation matrix again and it's
just going to swap the 3 and 2.

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00:13:50,840 --> 00:13:55,005
puts the vector back in the original order
1,2,3.