Transpose of a matrix
Transpose of a Matrix Product Taking the transpose of the product of two matrices
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- 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...
- This is the Jth column, so the Mth row is gonna look like this
- A m j. 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.
- So this entry right here
- let me just, I might
- I realize this might be useful
- this is going to be my nth row,
- this is 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 M by N matrix.
- And each of its rows become its columns
- and the same thing I did for A,
- its transpose is right there
- A was M by N, the transposes N by M
- and each of these rows become each of these columns.
- Now, fair enough.
- Let's define two new matrices right now.
- Let's define the matrix C.
- Let's define the matrix C.
- Let me do it over here where
- Let me do it over here
- I think...I think the real estate well be valuable
- in this video
- Let's define my matrix C as being equal to
- the product of A and B
- So what's...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...matrix product to be defined.
- And it's going to result
- with an M by N matrix.
- So it's going to be an M by M matrix.
- Now's let define another matrix,
- let's call it D. D...
- And it's equal to B tranpose
- 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.
- which is a requirement for this product to be defined
- and so the dimensions of B
- are going to be M by M. 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
- C one one, C two two, C one two...
- all the way to C one M.
- You can imagine it's cause its an M by N matrix
- you're going to have C M M over here
- you know how the...you know how this drill goes.
- But what I'm curious about is
- how do we figure out what C...the general C, i, j is?
- How do we figure out what a particular entry is?
- We know that C
- is the products of A and B.
- So to get to a particular entry in C...
- and we've seen this before
- so let's...
- a particular entry in C
- so C, i, j
- it's going to be
- you can view it as
- the dot product of the ith row in A
- the ith row in A
- with the jth column in B
- 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
- A, i,1...
- A, i, 1 times B, 1, j
- B, 1, j
- plus A, i, 2
- A, i, 2
- times B, 2, j...B, 2, j
- and you're just going to keep going
- until you get to the last term here
- A, i, n
- A, i, n
- times the last term here B, n, j.
- B, n, j.
- 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 know, you are going to have
- d 1 1, d 1 2
- sorry...d, 1, 2, all the way to
- d 1 m, you're going to have d m n
- I could keep putting entries here,
- but I'm curious about just the
- some general entry here
- let's say I want to find
- D sub j i
- D sub j i
- 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 the 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
- So we're going to take the dot product of the jth row here.
- Which is is that right there.
- With the ith column of A,
- with the ith column 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
- B, i, j
- well let me write it this way
- it's going to be B i j times
- A
- i
- one.
- Or we could write it as
- A i one times B one j.
- And it's going to be plus B 2 j times
- A i 2, which is
- the same thing as A i 2 times B 2 j.
- A i 2 times B 2 j.
- And you are going to keep going
- until you get B n j times A i n.
- Or, you could write that as
- A i n times B n j.
- B n j.
- Now, notice something.
- These two things are equivalent.
- They are completely equivalent statements.
- The...
- D sub j i is equivalent to C sub i j.
- Let me write that.
- So D....or I could write C sub i j,
- is equivalent to
- D sub j i.
- Or another way you could say it is
- anything that's at row...
- all that 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.
- True for all 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
- C transpose is equal to D.
- Or you could say that C is equal to
- D transpose.
- Now this is pretty interesting,
- cause how did we define these two?
- We said that our matrix C
- is equal to our matrice...
- our matrice...the matrix product A and B.
- And we said that D
- is equal to, our matrix product
- B transposed times A transposed.
- I did that...those definitions right there.
- Here are the definitions.
- 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
- transposed, is equal to D.
- So it is equal to D, which is just
- B transpose, A transpose.
- And this is a pretty neat takeaway.
- This is a pretty neat takeaway here.
- 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 reversed order.
- B transpose, A transpose.
- Which is a pretty pretty neat takeaway.
- And you can actually extend this to an arbitrary
- number of matrices that you are taking the
- product of. If you're taking...
- 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 a different...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 4 or 5
- 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 transposes equal to
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