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  • [inaudible] -- topic. I've been talking --

  • I've been multiplying matrices already, but certainly time for me

  • to discuss the rules for matrix multiplication.

  • And the interesting part is the many ways you can do it,

  • and they all give the same answer. So it's -- and they're

  • all important. So matrix multiplication,

  • and then, come inverses. So we're --

  • we -- mentioned the inverse of a matrix, but there's --

  • that's a big deal. Lots to do about inverses

  • and how to find them. Okay, so I'll begin with how to

  • multiply two matrices. First way,

  • okay, so suppose I have a matrix A multiplying a matrix B and --

  • giving me a result -- well, I could call it C.

  • A times B. Okay. So, l- let me just review the

  • rule for w- for this entry. That's the entry in

  • row I and column J. So that's the I J entry.

  • Right there is C I J. We always write the row number and

  • then the column number. So I might --

  • I might -- maybe I take it C three four, just to make it specific.

  • So instead of I J, let me use numbers. C three four.

  • So where does that come from, the three four entry?

  • It comes from row three, here, row three and column four,

  • as you know. Column four.

  • And can I just write down, or can we write down the formula for

  • it? C three four is --

  • if we look at the whole row and the whole

  • column, the quick way for me to say it is row three of A --

  • I could use a dot for dot product. I won't often use that,

  • actually. S- dot column four of B.

  • And -- but this gives us a chance to just, like, use a little

  • matrix notation. What are the entries?

  • What's this first entry in row three?

  • The first -- the first -- that number that's

  • sitting right there is A,

  • so it's got two indices and what are they? [Class response,

  • inaudible] Three one. So there's an A three one there.

  • Now what's the first guy at the top of column four? [Class

  • response, inaudible] So what's sitting up there?

  • [Class response, inaudible] B one four, right.

  • So that this dot product starts with A three one times B one four.

  • And then what's the next -- so this is like I-

  • I'm accumulating this sum, then comes the next guy, A three two,

  • second column, times B two four, second row.

  • So it's B A three two, B two four and so on.

  • Just practice with indices. Oh, let -- let me even practice with

  • a summation formula. So this is --

  • I -- I -- most of the course, I use whole vectors.

  • I very seldom, get down to the details of these

  • particular entries, but here we'd better do it.

  • So I'm -- it's some kind of a sright?

  • Of things in row three, column K shall I say?

  • Times things in row K, column four. Do you see that that's what's --

  • that's what we're seeing here? This is K is one,

  • here K is two, on along -- so up --

  • so the sum goes all the way along the row and down the column,

  • say, one to N. So that's what the A --

  • the C three four entry looks like. A sum of A three K B K four.

  • Just takes a little practice to do that.

  • Okay. And, uh -- oh, well,

  • maybe I should say -- when are we allowed to multiply

  • these matrices? What are the shapes of these things?

  • The shapes are, uh -- if we allow them to be not

  • necessarily square matrices. If they're square,

  • they've got to be the same size. If they're rectangular, they're

  • not the same size. If they're rectangular,

  • this might be -- well, I always think of A as M by N.

  • M rows, N columns. So that sum goes to N.

  • Now what's the point -- how many rows does B have to have?

  • [Class response, inaudible] N. N.

  • The number of rows in B, the number of guys that we meet

  • coming down has to match the number of ones across.

  • So B will have to be N by something. Whatever. P.

  • So then -- the number of columns here has to match the number of rows

  • there, and then what's the result? What's the shape of the result?

  • What's the shape of C, the output? Well, it's got these same M rows,

  • or -- it's got M rows. And how many columns?

  • [Class response, inaudible] P. M by P. Okay.

  • So there are M times P little numbers in there,

  • entries, and each one, looks like that.

  • Okay. So that's the standard rule. That's the way people think of

  • multiplying matrices. I do it too.

  • But that's -- that's -- I want to talk about other

  • ways to look at that same calculation,

  • looking at whole columns and whole rows.

  • Okay. So can I do A B C again? A B equaling C again?

  • But now, tell me s- tell me about, yeah.

  • Let me -- I'll put it up here. So here goes A, again,

  • times B producing C. And again,

  • this is M by N. This is N by P and this is M by P.

  • Okay. Now I want to look

  • at whole columns. I want to look at the columns of --

  • here -- in fact - here's the second way to multiply matrices.

  • Because I'm going to build on what I know already.

  • How do I multiply a matrix by a column?

  • How do I -- I know how to multiply this matrix

  • by that column. Th- th- shall I call

  • that column one? That tells me column

  • one of the answer. The matrix times the first column is

  • that first column. Because none of this stuff entered

  • that part of the answer. The matrix times the second column

  • is the second column of the answer. Do you see what I'm saying?

  • That I could think of multiplying a matrix by a vector,

  • which I already knew how to do, and I can think of

  • the vec- I can think of just P columns sitting side by side,

  • just like resting next to each other.

  • And I multiply A times each one of those.

  • And I get the P columns of the answer.

  • Do you see this as -- this is quite nice, to be able to

  • think, okay, matrix multiplication works so

  • that I can just think of having several columns,

  • multiplying by A and getting the columns of the answer.

  • So, like, here's column one -- a -- shall I call that --

  • here's a -- shall I call that column one?

  • And what's going in there is A times column one.

  • Okay. So that's the picture a column at a time.

  • So what does that tell me? What does that tell me

  • about these columns? These columns of C are combinations,

  • because we've seen that before, of columns of A.

  • Every one of these comes from A times this,

  • and A times a vector is a combination of the columns of A.

  • And -- right -- and it makes sense,

  • because the columns of A have length M and the columns of

  • C have length M. And every column of C is a --

  • is some combination of the columns of A.

  • And it's these numbers in here that tell me what combination it is.

  • Do you see that? That out --

  • that in that answer, C, I'm seeing stuff that's col-

  • that's combinations of these columns.

  • Now, suppose I look at it -- that's two ways now.

  • The third way is look at it by rows. So now let me change to rows. Okay.

  • So now I can think of a row of A -- a row of A multiplying all these

  • rows here and producing a row of the product.

  • So this row takes a combination of these rows and that's the answer.

  • So these rows of C are combinations of what?

  • Of -- tell me how to finish that. The rows of C,

  • when I have a matrix B, it's got it's rows and I multiply by

  • A, and what does that do? It mixes the rows up.

  • It makes -- it creates combinations of the rows

  • of -- [student response, inaudible] -- B, thanks.

  • Rows of B. That's what I wanted to see,

  • that this -- that this answer -- I can see where the pieces are

  • coming from. The rows in the answer are coming as

  • combinations of these rows. The columns in the answer are com-

  • coming as combinations of those columns.

  • And now that's -- so that's three ways.

  • Now you can say, okay, what's the fourth way?

  • The fourth way -- so that's -- now we've got,

  • like, the regular way, the column way, the row way and -- what's left?

  • The -- the one that I can -- I -- I want to tell you about --

  • well, one way is columns times rows. What happens if I multiply --

  • so th- this was row times column, it gave a number.

  • Okay. Now I want to ask you about column times row.

  • What does -- if I multiply a column of A times a

  • row of B, what shape am I ending up with?

  • So if I take a column times a row, that's definitely different from

  • taking a row times a column. So a column of A was --

  • what -- what's the shape of a column of A?

  • N by one. A column of A is a column.

  • It's got M entries and one column. And what's a row of B?

  • It's got one row and P columns. So what's the shape --

  • what do I get if I multiply a column by a row?

  • I get a big matrix. I get a full-sized matrix.

  • If I multiply a column by a row, I get -- should we just do one?

  • Let me take the column two three four times the row one six.

  • That is a -- that product there -- I mean, when I'm just following the

  • rules of matrix multiplication,

  • those rules are just looking like -- kind of

  • petite, kind of small, because the -- the rows here are so

  • short and the columns there are so short, but they're the same

  • length, one entry. So what's the answer?

  • What's the answer if I do two three four times one six,

  • just for practice? Well, what's the first row of the

  • answer? Two twelve. And the second row of the answer is

  • three eighteen. And the third row of the answer is

  • four twenty four. Actually, what am I --

  • I mean, that's a very special matrix, there.

  • Very special matrix. What can you tell me about its

  • columns, the columns of that matrix?

  • They're multiples of this guy, right?

  • They're multiples of that one. Which follows our rule.

  • We said that the columns of the answer were combinations,

  • but there's only -- to take a combination of one guy,

  • it's just a multiple. The rows of the answer,

  • what can you tell me about those three rows?

  • They're all multiples of this row. They're all multiples of

  • one six, as we expected. But I'm getting a full-sized

  • matrix. And now, just to complete this

  • thought, if I have,-- now, l- let me right

  • down the fourth way. A B is a sum of columns

  • of A times rows of B. So that, for example, if my --

  • if my matrix was two three four and then had another column,

  • say, seven eight nine, and my matrix here has --

  • say, started with one six and then had another column like zero zero,

  • then -- h- here's the fourth way, okay?

  • I've got two columns there, I've got two rows there.

  • So the beautiful rule is -- see, the whole thing by columns

  • and rows is that I can take the first column times the first

  • row and add the second column times the second row.

  • So that's the fourth way, that -- that I can take columns times rows,

  • first column times first row, second column times second row and add.

  • Actually, what will I get? What will the answer be for that

  • matrix multiplication? Well, this one it's just going to

  • give us zero, so in fact I'm back to this --

  • that's the answer, for that matrix multiplication. [sneeze]

  • Uh -- I'm -- I'm sort of,

  • like, happy to put up here these facts about matrix multiplication,

  • because it gives me a chance to write down special

  • matrices like this. This is a special matrix.

  • All those rows lie on the same line.

  • All those rows lie on the line through one six.

  • If I draw a picture of all these row vectors, they're all

  • the same direction. If I draw a picture of these two

  • column vectors, they're in the same direction.

  • Later, I would use this language. Not too much later, either.

  • I would say the row space, which is like all the combinations

  • of the rows, is just a line for this matrix.

  • The row space is the line through the vector one six.

  • All the rows lie on that line. And the column space is also a line.

  • All the columns lie on the line through the vector two three four.

  • So this is like a really minimal matrix.

  • And it's because of these ones. Okay. So that's a third way.

  • Now, even -- yeI -- can I --

  • will you -- would you take -- this is -- I- I want to say one more

  • thing about matrix multiplication while we're on the subject.

  • And it's this. You could also multiply --

  • you could also cut the matrix into blocks and do the multiplication

  • by blocks. Yethat's actually so,

  • useful that I want to mention it. Block multiplication.

  • So I could take my matrix A and I could

  • chop it up, like, maybe just for simplicity,

  • let me chop it into two -- into four square blocks.

  • Suppose it's square. Let's just take a nice case.

  • And B, suppose it's square also, same size.

  • So these sizes don't have to be the same.

  • What they have to do is match properly.

  • Here they certainly will match. So here's the rule for

  • block multiplication, that if this has blocks like,

  • A -- so maybe A one, A two, A three, A four

  • are the blocks here, and these blocks are B one,

  • B two, B three and B four? Then the answer I can find block --

  • I can find that block. And if you tell me what's in that

  • block, then I'm going to be quiet about matrix multiplication for the

  • rest of the day. What goes into that block?

  • You see, these might be -- this matrix might be --

  • these matrices might be, like, twenty by twenty with blocks that

  • are ten by ten, to take the easy case where all the

  • blocks are the same shape. And the point is that I could

  • multiply those by blocks. And what goes in here? What's that?

  • What's that -- what's that block in the answer?

  • A one B one, that's a matrix times a matrix,