So that's the picture a column at a time. So, like, here's column one shall I call that column one?Īnd what's going in there is A times column one.
That I could think of multiplying a matrix by a vector, which I already knew how to do, and I can think of just P columns sitting side by side, just like resting next to each other.Īnd I multiply A times each one of those.ĭo 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. The matrix times the second column is the second column of the answer. The matrix times the first column is that first column.īecause none of this stuff entered that part of the answer. I know how to multiply this matrix by that column. I want to look at the columns of - here's the second way to multiply matrices.īecause I'm going to build on what I know already. So here goes A, again, times B producing C. That's the way people think of multiplying matrices.īut I want to talk about other ways to look at that same calculation, looking at whole columns and whole rows. So there are m times P little numbers in there, entries, and each one, looks like that. Well, it's got these same m rows - it's got m rows. So the number of columns here has to match the number of rows there, and then what's the result? The number of rows in B, the number of guys that we meet coming down has to match the number of ones across. Now what's the point - how many rows does B have to have?
If they're rectangular, this might be - well, I always think of A as m by n. If they're rectangular, they're not the same size. If they're square, they've got to be the same The shapes are - if we allow them to be not necessarily square matrices. And - well, maybe I should say - when are we allowed to multiply these matrices? So that's what the C three four entry looks like. This is K is one, here K is two, on along - so the sum goes all the way along the row and down the column, say, one to N. Of things in row three, column K shall I say?ĭo you see that that's what we're seeing here? I very seldom, get down to the details of these particular entries, but here we'd better do it. So this is - most of the course, I use whole vectors. Oh, let me even practice with a summation formula. And then what's the next - so this is like I'm accumulating this sum, then comes the next guy, A 3 2, second column, times B 2 4, second row. So that this dot product starts with A 3 1 times B 1 4. Now what's the first guy at the top of column four? That number that's sitting right there is.Ī, so it's got two indices and what are they?
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.īut this gives us a chance to just, like, use a little matrix notation. It comes from row three, here, row three and column four, as you know.Īnd can I just write down, or can we write down the formula for it? So where does that come from, the three four entry? So I might - I might - maybe I take it C 3 4, just to make it specific.Ĭ 3 4. We always write the row number and then the column number. So, let me just review the rule for this entry. Okay, so I'll begin with how to multiply two matrices.įirst way, okay, so suppose I have a matrix A multiplying a matrix B and - giving me a result - well, I could call it C. Lots to do about inverses and how to find them. So matrix multiplication, and then, come inverses. I've been multiplying matrices already, but certainly time for me to discuss the rules for matrix multiplication.Īnd the interesting part is the many ways you can do it, and they all give the same answer.