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Topic five is Matrix Multiplication.

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So, I think we're developing some idea of
what it means to multiply a vector by a

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matrix, and now we'll just look at how we

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can extend that idea to multiplying two
matrices together.

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So we have a linear system Ax equals b.

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And now we know how to make these
elimination matrixes, matrices E.

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So, I can make and eliminate, an
elimination matrix E.

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And I know what that's supposed to do.

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So, I want to define matrix
multiplication,

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so that the product of E and my
coefficient matrix.

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Gives me the same coefficient matrix I
would get if I did one elimination step.

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So we know how to do the vector.

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So I know how to do elimination on the
right hand side now.

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And I know what effect elimination is
going to

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have on the coefficient matrix so I
want to make

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sure that whatever definition I come up
with

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for matrix multiplication it, it needs to
do that.

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So for our.
Column view.

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I, I think of the matrix as composed of n
columns.

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So here, a1, a2 up to an, these are the
columns of the matrix A.

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And I know how to do vector times alright
sorry, matrix times vector multiplication.

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So

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I could think of one way I might want to
represent matrix multiplication is just,

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I'll say the answer needs to be E times
the first column of A, E

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times the second column of A, and then
each one of those is going to

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be a vector, and I'm going to put all
those vectors back together in a matrix.

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And so this would then be the answer

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to my matrix multiplication product, so my
matrix

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product, E times A.

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Okay.

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So we have to, to make this a little bit
more rigorous,

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we have to start thinking of exactly what
rules we have to follow.

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So a matrix is just a rectangular array of
numbers.

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If we say an m by n matrix, we mean that
it has m rows and n columns.

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And I'm going to denote the entries in the
matrix by little aij.

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So if it has, I'm sort of using the
notation here if it's got one subscript.

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So if I say just A sub j that's a whole
column.

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If it's A sub i that's a row.

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Or if it's aij if it has two

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subscripts then that's a particular entry
in a matrix.

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And so that means I can write my matrix A
like this.

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So I have a11 is always the top left entry
and then there's a21 below that, a31 below

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that all the way down to am1 and then for
columns I have a11.

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Just one step to the right of that is a12,
a13, and so on up to a1n.

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And then the bottom right corner will just
be amn.

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So this works well for small, smaller
matrices.

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I suppose if you wanted to actually start
talking about you know, the.

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13, 17 position, you might want to put a
comma in between here,

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[COUGH]

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and so for addition, we're

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going to do addition for matrices

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just like we did with vectors, and so it's
going to be For

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vectors I call the elements components and
for a matrix I'll call the

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elements entries, like that.
So if matrices,

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two matrices are the same dimension, so
they have the same number

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of rows and the same number of columns,
then I can add them.

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And I'm going to say that the sum is just
The element-wise sum.

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And then I can also multiply a matrix by a
scalar value, C.

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And that's going to be the same as the
vector.

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It's just going to be each entry in the
matrix multiplied by that value, C.

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So, for instance, if I wanted to add these
two matrices together, I have in the top

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left, so the 1 1 sum is just going to be 1
plus 2, so I get 3.

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The top right entry is going to be 2 plus
2, 4.

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And, you know, every other one.

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So this would be 0 plus 9 is equal to 9.

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And then if I want to multiply by a scalar
value C.

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So here I decided to choose 2 as my scalar
value.

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Then that's going to have the affect of
just

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doubling every element or every entry in
the matrix.

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So 1,2 becomes 2,4.

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3,4 becomes 6,8 and then 0,0 stays 0,0.

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So addition is pretty straightforward,
scaling by a

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[NOISE]

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A real valued scalier is pretty straight
forward.

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Matrix multiplication we're going to find
is a little bit more difficult.

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So in order to plot, in order to multiply
a matrix A by a matrix

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B the number of columns of A has to be
equal to the number of rows of B.

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So in particular if A is an m by n matrix,
and B is an n by p

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matrix, then I'm going to have m rows and
n columns for A.

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And then the number of columns of A has to
be equal to the number

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of rows, so I have to have n columns in A
and n rows in B.

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If that's true, then I can make a matrix
product.

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And the output is going to be the
dimension of

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the output, so the dimension of the
product is going

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to ha have a number of rows determined by
the first.

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Element in the product, and a number of
columns determined by the second.

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So the product AB is going to have m rows
because A had

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m rows and it's going to have p columns
because B had p columns.

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And so one thing you can think of is the
matrix

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form of the dot product is just the most
extreme case.

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So, if I have two vectors, in this case,

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of length two, and I take their dot
product.

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I could also write that as u transpose, so
that means,

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transpose of a vector just means turn it
on it's side.

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So, here u is a column vector.
When I transpose it.

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It becomes a row.

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So I now have a matrix that has one row
and

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two columns, times a matrix with two rows
and one column,

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and that's just going to be the, the
standard dot product.

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And so what was important here was that
the number of columns,

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so in this case two, And the number of
rows that had to

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match and then I'm able to make this
product, so this is, this

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provides a very easy check whenever you're
doing any sort of matrix computation.

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you know if the dimensions of your
matrices don't fit in

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your products then you've done something
wrong.

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And so then the rule to find the actual
entries in the product AB.

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So we can say this is going to have the
dot products, so I

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think this is the easiest way, if you, if
I'm just interested in calculating.

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The product of two matrices.

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This is the easiest way to think about it.

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So, the entries in the matrix AB are the
dot

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products of the rows of A and the columns
of B.

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So I'm using this notation.
So AB is the, a matrix product.

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And then I

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want to take the ith, jth element of that.

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And the way I'm going to get that, so
that's going to be one single number.

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And remember dot products give me a scaler
value which is one single number.

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The way I'm going to figure out what that
number is, is I'm going to take the

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ith row of a, and take the dot product of
that and the jth column of B.

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And then, once I know what the dimensions
are,

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I'll know what the ranges of i and j are.

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And then I just have to fill in the matrix
one by one.

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You can also write this out using a sum
notation.

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So, here, just instead of writing AB.

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I'm going to call that matrix C, so C is
the product of the matrix A and B and then

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C sub ij are the entries of that matrix
and this is basically just

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writing out exactly the same operation
that this would be doing, so here,

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I'm summing over k, so k is the thing
that's changing.

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So, this i here is the same for all n of
the, the pieces of this

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sum.
And similarly, j is staying the same.

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So that means that the row of the matrix A
that I'm looking at.

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Is determined just by the position of C in

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the, in the answer, and then I'm going
across the row and

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down the column to compute the actual
value for that,for that entry C.

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And so, for example, you can just do a
quick two by two example, so If

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I want to know where this 5 came from,
this 5 is in the 1 1 position.

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So that tells me it's the dot product of
the

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first row of A, and the first column of B.

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So that would be 1 times 2, plus 1 times
3, which gives me 5.

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Or I can look at Cij, so C

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1 1.
So this number here would be C, 1, 1.

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Would be a, 1, 1 time b, 1,

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1 plus a, 1, 2 times b, 1, 2, which is
exactly the, the same thing.

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Ops, sorry.
B, 2, 1.

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So a 1 1, b 1 1, a 2 1, sorry,

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a 1 2, b 2 1, which gives me the same dot
product.

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So I find it tricky to keep track of all
these subscripts,

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as you just witnessed, so I prefer
thinking of it like this.

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Or, more likely, using a computer to
calculate these things for me.

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And one thing to keep in mind, so, for two
by

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two and three by three cases, generally it
seems pretty straightforward.

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But, really what your aiming at is solving
much bigger systems of equations.

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So you want to think about how much work
it's actually going to, going to take.

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So, you know, now again you have a
computer, so it's not so bad.

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At least it's not you doing the work.

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But imagine if you had to solve a problem
on a chalk board.

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You know, you would have really

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two considerations.

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One is going to be, how you know, how

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many different operations am I going to
need to do?

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So, for each cell in this matrix, so this
one is, if

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it's a two by two matrix, there's going to
be four cells

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[COUGH]

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I need to do n multiplications, so that's
just my dot product, and then to add those

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up, I need to do n minus 1 additions.
And then there's going to be,

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so in this case, it's square, so it's n
times n,

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so this algorithm, I'm going to have to do
something that takes.

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2n roughly steps, and I'm going to have to
do that n times n times, so it's going to

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take 2 times n cubed mathematical
operations, like simple

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mathematical operations, for me to do this
matrix product.

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And so the other consideration you might
want to think is

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how much memory is that going to take.

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So if I was doing this on a chalkboard or

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a piece of paper the chalkboard's nice
because I can

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actually reuse parts of it by erasing a
number I

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don't need anymore and writing another
number in that space.

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It's a little more difficult to do with a
piece of paper.

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surprisingly if you are programming a
computer, generally when you

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want to optimize an algorithm, you're
doing pretty much the same thing.

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You want to use the least amount

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of memory and do the least amount of
mathematics, so.

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A lot of what was developed initially when
people were trying to solve

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these problems by hand, is still really

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quite useful for solving them with
computers.

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Okay, so an inner product is a row times a
column.

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So this was the, the example I gave of

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a dot product being a simple example of
matrix multiplication.

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So, if I have a row times a column then
there's only going to be, so that's

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one row with several columns times one
column with several rows.

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So, the answer is going to be a scaler.

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I can actually move those and do, or do
that in the opposite order.

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So now I can do a single col, so a matrix

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that's a single column times a matrix
that's a single row.

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And that gives me something called an
outer product

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and this outer product is now going to be

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going to be a matrix so I have a three by
one matrix times a one by three matrix.

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And so it's that three on the outside and,
the,

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so at the beginning, and the three at the

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end that give me the dimension of the
answer.

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And then the ones that have to match up on

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the inside so I can actually do the matrix
multiplication.

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So, inner product gives me a scalar value.

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An outer product gives me a matrix like
this.

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And some useful facts about matrix
multiplication so each column

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of AB is a linear combination of the
columns of A.

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So if I look at A times the column of

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B this is exactly the same thing I was
doing.

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When I was multiplying a matrix times a
vector, and I said that I could

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interpret that using my column picture as
a linear combination of the columns of A.

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And so

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that gives me each column of the product
is going

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to be a linear combination of the columns
of A.

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And then I can get a similar result So the
rows of the product, so the row i of

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AB, the rows of the product are going to
be linear combinations of the rows of B.

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So here, this is, this is sort of how I
want to think, think about it.

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I take one row of i and that's, or sorry,
one row of A.

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And that's going to give me the
coefficients for the rows of B.

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They give me row i of AB.
And so then laws

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for matrix operations.

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We get a commutative law.

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So for addition we have A plus B is equal
to B plus A.

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And you can show that with just the same
arguments

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00:15:50,230 --> 00:15:53,060
that I did for vectors because it's
defined entry wise.

224
00:15:54,680 --> 00:15:58,250
I get a distributive law, so if I take the
sum

225
00:15:58,250 --> 00:16:03,130
of A plus B and scale that by scaler value
c.

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00:16:03,130 --> 00:16:08,390
Then that's equal to A scaled by that
constant and plus B

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00:16:08,390 --> 00:16:09,660
scaled by the same constant.

228
00:16:11,560 --> 00:16:17,280
I get an associative law for addition, so
it doesn't matter if I want to add A+B+C,

229
00:16:17,280 --> 00:16:22,180
I can add B+C first or I can add A+B
first, I'm going to get the same answer.

230
00:16:24,300 --> 00:16:27,660
For multiplication it's a little bit
trickier.

231
00:16:27,660 --> 00:16:33,190
So I get left distributive law so if I
have C times

232
00:16:33,190 --> 00:16:38,100
A plus B that's equal to C times A plus C
times B.

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00:16:40,690 --> 00:16:46,420
I get a right distributive law So A plus B
times C

234
00:16:46,420 --> 00:16:50,560
is equal to AC plus BC and the important
thing ops the

235
00:16:50,560 --> 00:16:56,190
important thing to keep, keep track of
here if is C comes

236
00:16:56,190 --> 00:17:00,620
first, it has to always come first when
I'm doing this distributive operation.

237
00:17:01,690 --> 00:17:04,000
If the A and B come first.

238
00:17:04,000 --> 00:17:05,820
Then the A and B have to

239
00:17:05,820 --> 00:17:08,180
come first when I'm doing this
distributive operation.

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00:17:11,180 --> 00:17:15,000
And then I also get an associative law, so
if I want to calculate the product of

241
00:17:15,000 --> 00:17:20,300
three matrices, A plus, or, A time B times
C, they have to be the right dimensions.

242
00:17:20,300 --> 00:17:25,420
So The number of columns of A has to equal
the number of rows of B.

243
00:17:25,420 --> 00:17:27,920
And the number of rows of B has to equal
the number of.

244
00:17:29,450 --> 00:17:29,630
Sorry.

245
00:17:29,630 --> 00:17:32,670
The number of columns of B has to equal
the number of rows of C.

246
00:17:34,090 --> 00:17:35,130
I think I got that right.

247
00:17:36,522 --> 00:17:42,340
and, but if I can do those operations then
it doesn't matter what order I do them in.

248
00:17:42,340 --> 00:17:46,750
So, I can do The product BC first and then
multiply that by

249
00:17:46,750 --> 00:17:50,550
A or I can do the product AB first and
multiply that by C.

250
00:17:54,000 --> 00:17:59,530
So one law we don't get is a commutative
law for matrix multiplication

251
00:17:59,530 --> 00:18:05,650
so almost all the time A times B is not
even equal to B times A.

252
00:18:05,650 --> 00:18:06,780
And it's even worse than that.

253
00:18:07,840 --> 00:18:13,450
So, if p is not equal to m, so remember A
is an m by n matrix, and

254
00:18:13,450 --> 00:18:19,240
B is an n by p matrix, if p is not equal
to m, then I can't

255
00:18:19,240 --> 00:18:21,610
even do this operation, because the
dimensions

256
00:18:21,610 --> 00:18:23,360
won't line up when I switch the order.

257
00:18:26,690 --> 00:18:31,830
If both A and B are m by n matrices,
sorry,

258
00:18:31,830 --> 00:18:36,970
if A is m by n and B is n by m, then

259
00:18:36,970 --> 00:18:41,939
if I do the product AB I'm going to end up
using these outer dimensions.

260
00:18:44,370 --> 00:18:49,440
When I do the product BA, now the outer
dimensions are going to be n and n.

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00:18:49,440 --> 00:18:54,960
So AB is an m by m matrix.
BA is an n by n matrix.

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00:18:54,960 --> 00:18:56,570
So these things, they can't be equal

263
00:18:56,570 --> 00:18:58,670
to one another if they have different
dimensions.

264
00:18:59,860 --> 00:19:02,060
So, for equality, they have to be the same

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00:19:02,060 --> 00:19:04,540
dimension and then each entry has to be
equal.

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00:19:09,520 --> 00:19:13,510
And then when we have square matrices, so
if we have n by n matrices, at

267
00:19:13,510 --> 00:19:17,970
least I can do the products and they'll be
the right size.

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00:19:17,970 --> 00:19:23,782
But suppose I let A equal 0,0,1,0

269
00:19:23,782 --> 00:19:29,666
and B equal 0,1,0,0.
So if I do this product AB ops- product

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00:19:29,666 --> 00:19:35,068
AB that gives me 0 0 0 1.
But if I do the product BA, I get

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00:19:35,068 --> 00:19:36,690
1 0 0 1.

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00:19:36,690 --> 00:19:41,730
So even when I can do it and they're the
right size, you're still not

273
00:19:41,730 --> 00:19:45,990
guaranteed to get the same answer if I do
the multiplication in the opposite order.

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00:19:51,130 --> 00:19:53,200
So one exception to this rule,

275
00:19:55,870 --> 00:20:01,270
so this is a constant c, a scalar value.
And this

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00:20:01,270 --> 00:20:06,480
is the identity matrix, yeah.
It's kind of hard to see with this font.

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00:20:06,480 --> 00:20:12,350
So, one matrix that always Commutes is the
identity matrix.

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00:20:12,350 --> 00:20:15,880
And so, if it's scaled by a constant
value, that will also commute.

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00:20:15,880 --> 00:20:21,200
So, if you have a identity matrix times
something, and they're square,

280
00:20:21,200 --> 00:20:23,360
then you can switch the order of that
multiplication.

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00:20:23,360 --> 00:20:23,860
And

282
00:20:27,960 --> 00:20:30,660
then one more thing that I want to
mention, just because

283
00:20:30,660 --> 00:20:34,500
I don't know really where else it fits in,
matrix powers.

284
00:20:34,500 --> 00:20:36,920
So a power of a matrix if I tell, talk

285
00:20:36,920 --> 00:20:40,620
about A squared, then A needs to have the
right dimension.

286
00:20:40,620 --> 00:20:42,329
So basically it needs to be a square
matrix.

287
00:20:45,860 --> 00:20:51,150
if I have A squared times say A cubed
that's equal

288
00:20:51,150 --> 00:20:56,220
to A to the fifth power so these behave
just like numbers would and if

289
00:20:56,220 --> 00:21:01,350
I have say A squared and that quantity
cubed that's equal to A to the sixth.

290
00:21:03,110 --> 00:21:07,520
And then if I talk about A to the 0,
that's going to be equal to 1.

291
00:21:07,520 --> 00:21:11,680
And 1, when I'm talking about matrices is
the identity matrix.

292
00:21:11,680 --> 00:21:14,660
>> Did you say that A had to be a square
matrix for that to work?

293
00:21:14,660 --> 00:21:16,210
Because I, I was writing down.

294
00:21:16,210 --> 00:21:19,840
>> Well because you can only multiply A
by A, if it has the

295
00:21:19,840 --> 00:21:22,740
same number of rows and columns, so that
means it has to be square.

296
00:21:24,530 --> 00:21:24,760
Okay.

297
00:21:24,760 --> 00:21:25,260
[COUGH]

298
00:21:28,110 --> 00:21:32,820
Then, I can also do something called
putting a matrix into block form.

299
00:21:33,820 --> 00:21:36,670
So here I have a matrix with four rows and
six columns.

300
00:21:37,730 --> 00:21:41,130
And then I'm going to break it into six
blocks, and

301
00:21:41,130 --> 00:21:44,460
each block is going to be a two by two
identity matrix.

302
00:21:44,460 --> 00:21:47,700
So I could also write A like this, so
these

303
00:21:47,700 --> 00:21:52,850
are just six identity matrices, they're
all two by two.

304
00:21:56,980 --> 00:22:01,470
And then if I make these blocks the
correct size, I can do.

305
00:22:01,470 --> 00:22:04,160
Block multiplication following the same
rules

306
00:22:04,160 --> 00:22:06,360
that I would do for matrix multiplication.

307
00:22:06,360 --> 00:22:11,240
So if I have a matrix, A, that I can split
into four blocks like this,

308
00:22:12,830 --> 00:22:17,350
then when I want to figure out what the
product of this matrix is, I can do

309
00:22:17,350 --> 00:22:22,140
the, the product block-wise.
So I just was Just pretend like these

310
00:22:22,140 --> 00:22:23,100
are values.

311
00:22:23,100 --> 00:22:28,480
It would be A11 times B11 plus A12 times

312
00:22:28,480 --> 00:22:31,300
B21 which is exactly what I end up with
here.

313
00:22:31,300 --> 00:22:35,960
So, it's the same formula for matrices,
except now I'm talking about blocks

314
00:22:35,960 --> 00:22:41,139
of matrices, or sorry, blocks, which are
kind of sub matrices of a matrix.

315
00:22:43,180 --> 00:22:45,380
And the only caveat here is you have to
define

316
00:22:45,380 --> 00:22:48,540
your blocks so that all of these
operations are permitted.

317
00:22:52,010 --> 00:22:55,490
And the reason I want to do this is it
lets me do one last

318
00:22:57,580 --> 00:23:03,570
clever thing, which I can also now define
matrix multiplication in

319
00:23:03,570 --> 00:23:07,500
terms of columns, so I can get a sort of
column space.

320
00:23:08,600 --> 00:23:15,780
So my first definition was rows of A times
columns of B, if I make my blocks the

321
00:23:15,780 --> 00:23:22,009
columns of A and the rows of B and then I
use this rule here, what I end up with

322
00:23:22,009 --> 00:23:22,509
[INAUDIBLE]

323
00:23:25,580 --> 00:23:31,470
Is a column of A, times a row of B.
But if I have

324
00:23:31,470 --> 00:23:37,370
a column times a row, that was an outer
product, so that's actually a matrix.

325
00:23:37,370 --> 00:23:43,560
And it turns out that, what I end up with,
are n matrices.

326
00:23:43,560 --> 00:23:48,790
So here, ai, bi, is an outer product, so
that's a matrix.

327
00:23:48,790 --> 00:23:50,490
I have n matrices like this,

328
00:23:50,490 --> 00:23:55,380
one from each row-column, or, column-row
combination here.

329
00:23:55,380 --> 00:24:00,260
And then I add up those n matrices, and
that will also give me this product, AB.