Transpose of a matrix product

I've got a handful of matrices here. I have the matrix A that's an m by n matrix. You can see it has n columns and m rows. And actually let me throw in one entry there. It might be useful. This is the jth column. So the row is going to look like this-- amj. That's that entry right there. And then I have matrix B defined similarly, but instead of being an m by n matrix, B is an n by m matrix. And so this entry right here. Let me just-- I realize this might be useful. This is going to be my nth row. And it's going to be my jth column. And then I also wrote out their transposes. So if you look at the transpose of B, B was an n by m matrix. Now the transpose is going to be an m by n matrix. And each of its rows become its columns. And the same thing I did for A. It's transpose is right there, A was m by n. The transpose is n by m. And each of these rows because each of these columns. Now fair enough. Let's define two new matrices right now. Let's define the matrix C. Let me do it over here. I think the real estate will be valuable in this video. Let's define my matrix C as being equal to the product of A and B. So what are going to be the dimensions of C? Well, an m by n matrix times an n by m matrix, these two have to be equal even for the matrix product to be defined. And that's going to result with an m by m matrix. Now let's define another matrix. Let's call it D. And it's equal to B transpose times A transpose. And the dimensions are going to be the same, because this is an m by n times an n by m. So these are the same. So which is a requirement for this product to be defined. And so the dimensions of B are going to be m by m. So let's explore a little bit what the different entries of C are going to look like. So let me write my matrix C right here. So it's just going to have a bunch of entries-- c11, c12, all the way to c1m. You can imagine because it's an m by n matrix, you're going to have cmm over here. You know how this drill goes. But what I'm curious about is how do we figure out what the general cij is? How do we figure out what a particular entry is? We know that C is the product of A and B. So to get to a particular entry in C-- and we've seen this before-- so cij-- It's going to be-- you can view it as the dot product of the ith row in A with the jth column in B, just like that. And what's that going to be equal to? It's going to be equal to ai1 times b1j plus ai2 times b2j. And you're just going to keep going until you get to the last term here, ain times the last term here, bnj. Fair enough. Now what about our matrix D? What are its entries going to look like? So D, similarly, it's going to look like-- you're going to have d11, d12, all the way to d1m. You're going to have dmm. I could keep putting entries here. But I'm curious about just that sum in general entry here. Let's say I want to find d sub ji. That's what I want to find. So I want to find a general way for any particular entry of d. The jth row and ith column, which is a little bit different than the convention we normally use for these letters. But it's fine. The first one is D's row. The second one is D's-- is this entry's column. So how do we figure that out? So d sub j i. It's going to be equal to-- D is the product of these two guys. So to get the jth row and ith column entry here, we essentially take the dot product of the jth row here, which is that right there, with the ith column of A, which is that right there. So I'm going to take the dot product of that. And you might already see something interesting here. This thing right here is equivalent to that thing right there. And this thing right here is equivalent to that thing right there, because we took the transposes. But let's actually just write it out. So what is this dot product going to be equal to? Well, it's going to be bij. Let me write it this way. It's going to bij times ai1. Or we could write it as ai1 times b1j. And it's going to be plus b2j times ai2, which is the same thing as ai2 times b2j. And you're going to keep going until you get b and j times ain. Or you could write that as ain times bnj. Now notice something. These two things are equivalent. They're completely equivalent statement. The d sub ji is equivalent to c sub ij. Let me write that. Or I could write c sub ij is equivalent to d sub ji. Or another way you could say it is, all the entries that's at row i, column j in C is now in row j, column i in D. And this is true for all the entries. I stayed as general as possible. So what does this mean? This is the definition of a transpose. So we now get that C transpose is equal to D. Or you could say that C is equal to D transpose. Now this is pretty interesting, because how did we define these two? We said that our matrix C is equal to the matrix product A and B. And we said that D is equal to our matrix product B transpose times A transpose. I did those definitions right there. Here are the definitions. And now we just found out that D is equal to the transpose of C. So we could write that C transpose, which is the same thing as A times B transpose, is equal to D. So it is equal to D, which is just B transpose A transpose. And this is a pretty neat takeaway. If I take the product of two matrices, and then transpose it, it's equivalent to switching the order, or transposing them, and then taking the product of the reverse order-- B transpose, A transpose-- which is a pretty, pretty neat take away. And you could actually extend this to an arbitrary number of matrices that you're taking the product of. I'm not proving it here, but it's actually a very simple extension from this right now. If you take the matrices, let's say A-- let me do different letters-- X, Y, Z, if you take their product and then transpose it, it's equal to Z transpose, Y transpose, X transpose. I haven't proven this general case, and you could keep doing it with four or five or n matrices multiplied by each other, but it generally works. And you could essentially prove it using what we proved in this video right here, that you take the product of two matrices, take their transpose, it's equal to the product of their transposes in reverse order.