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djz

Why is matrix multiplication called that?

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Is it just me, or is the add+sum row by column matrix multiplication system a little contrived? It's obviously useful, but I don't understand why the process of adding and summing is called 'multiplication', and not something else. If I were to intuitively attempt to solve:
|x,2,1||2,1,x|
|3,1,2||3,3,y|
|4,y,z||1,2,3|
If was naive about the correct way to multiply matrices, and knowing that we can multiply by a scalar like so..
 |1,0,1|   |1x,0x,1x|
x|2,1,2| = |2x,1x,2x|
 |4,1,1|   |4x,1x,1x|
If I were asked to 'multiply' the two matrices, I would naturally come up with
|x,2,1||2,1,x|   |2x,2,1x|
|3,1,2||3,3,y| = |9, 3,2y|
|4,y,z||1,2,3|   |4,2y,3z|
As opposed to:
| 2x+7,    x+8,     , x^2+2y+3   |
|   11,     10,     , 3x+(y+6)   |
|3y+(z+8), 3y+(2z+4), 4x+(y^2+3z)|
Is there a name for this? Why is the summing of the products of row 1 of A + col 1 of B called 'multiplication' as opposed to the operation of multiplying A(x,x) by B(x,x)?

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First off. It might be easier to think of the multiplication in terms of vector*matrix as opposed to scalar*matrix.
Since that is 3 dot products. Using a matrix, you are just multiplying 3 vectors by one matrix to come up with 3 new vectors for your matrix.

But why is it multiplication? probably because it has similar properties. You can design a matrix that will result in AxB=>C behaving like n*2=>2n. You can reverse a multiplication using division. Unlike regular multiplication you can't change the side (non-communitive) but like regular multiplication you can change order(associative). And if you multiply out a 1x1 matrix with a 1x1 matrix, you get regular multiplication. Its really just an extension of the rules for a multiple variable(dimension) system.

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I think it mostly has to do with the fact that matrix multiplication is used as an abstraction for systems of equations, e.g.,


2x + 3y + 3 = 0
4x - y - 2 = 3
+ 2y - 3 = 1


becomes


[2 3 3][ x ] [ 0 ]
[4 -1 -2][ y ] = [ 3 ]
[0 2 -3][ 1 ] [ 1 ]


And this was generalized further to matrix*matrix instead of just matrix*vector

Additionally, it was probably named multiplication because it exhibits properties similar to scalar multiplication (associativity, among other things)

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Quote:
Original post by djz
Is there a name for this?
Element-wise multiplication.
Quote:
Why is the summing of the products of row 1 of A + col 1 of B called 'multiplication' as opposed to the operation of multiplying A(x,x) by B(x,x)?
For the same reason 42 * 23 = 966 rather than 86: Because it's useful for what the quantities represent. If you want the deeper answer, try this.

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Matrices are often used to represent a linear transformation. (A linear transformation is a function which so that F(a*V + b*U) = a*F(V) + b*F(U), where U and V are vectors and a,b scalars.) For 1-dimensional transforms, we drop the matrix and vector notation since this is just one number, and we get equations like F(x) = a*x, with a being a scalar. For 2-dimensions and higher, we get F(V) = M*V, where M is a matrix and V a vector. In other words, Matrix multiplication is a higher-dimensional extension of multiplication. Hence it's name, among other reasons.

What you show as "naive multiplication" might be called component-wise multiplication. As far as I know, it has little practical or theoretical use in comparison to matrix multiplication which has a wide rang of uses.

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Mathematically speaking there is a reason for calling it multiplication.

In group theory a group is defined as a set which has a binary operation that satisfies certain criteria

Associativity: a*(b*c) = (a*b)*c
Closure: if a and b are in the set then a*b is also in the set.
Identity element: There has to be an element in the group, usually denoted 1 for which 1*a = a*1 = a for every a in the group.
Inverses: Each element has to have an inverse element for which a*inverse_of_a = 1 (and the inverse has to be in the group)

Now, if you aren't going to study at least uni level maths then that is irrelevant to your life. However, you can probably see that the real numbers (minus zero) form a group if you say that the binary operation is multiplication.

Then with a bit more thinking you can probably work out that various subsets of the set of all matrices with dimension n also form groups.

Anyway, that's a lot of rambling but the point is that the name multiplication comes from the idea that it is the group binary operation and addition was already taken.

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davetyler's post is almost correct. The more relevant algebraic structure that square matrices fit in is ring, which is a set where you can add, subtract and multiply. The operation we normally call "matrix multiplication" indeed satisfies the ring axioms.

However, multiplication is also defined for non-square matrices. It still makes sense to use this name in the general case, because this gives you a very convenient notation to do computations, and common manipulations still work (e.g., AB+AC=A(B+C)). You just have to be careful with the fact that matrix multiplication is not commutative (i.e., AB=BA is false in general).

In the end, it's only a matter of convention, and in this case, it's a useful one. As has been mentioned before, matrices are often used to represent linear transformations, and multiplication corresponds to composition. The multiplication that you propose doesn't have any interesting interpretation when you think of matrices as linear transformations, and its results would be different if you used different bases for your computations.

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Quote:
Original post by alvaro
davetyler's post is almost correct. The more relevant algebraic structure that square matrices fit in is ring, which is a set where you can add, subtract and multiply. The operation we normally call "matrix multiplication" indeed satisfies the ring axioms.

However, multiplication is also defined for non-square matrices. It still makes sense to use this name in the general case, because this gives you a very convenient notation to do computations, and common manipulations still work (e.g., AB+AC=A(B+C)). You just have to be careful with the fact that matrix multiplication is not commutative (i.e., AB=BA is false in general).

In the end, it's only a matter of convention, and in this case, it's a useful one. As has been mentioned before, matrices are often used to represent linear transformations, and multiplication corresponds to composition. The multiplication that you propose doesn't have any interesting interpretation when you think of matrices as linear transformations, and its results would be different if you used different bases for your computations.


Not aggressive or argumentative but I was wondering what in particular was incorrect in my post?

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Quote:
Original post by davetyler
Not aggressive or argumentative but I was wondering what in particular was incorrect in my post?


Hmmm... I didn't word that very well, did I? I guess I sounded both aggressive and argumentative. :)

I meant to say that multiplication wasn't chosen just because "addition was already taken". In fact, some of the axioms of ring involve both addition and multiplication, and they play together nicely. There is nothing wrong with your Math.

Sorry about the lack of nuance in my previous post. Can I still play the "I'm not a native English speaker" card?

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Quote:
Original post by alvaro
Quote:
Original post by davetyler
Not aggressive or argumentative but I was wondering what in particular was incorrect in my post?


Hmmm... I didn't word that very well, did I? I guess I sounded both aggressive and argumentative. :)

I meant to say that multiplication wasn't chosen just because "addition was already taken". In fact, some of the axioms of ring involve both addition and multiplication, and they play together nicely. There is nothing wrong with your Math.

Sorry about the lack of nuance in my previous post. Can I still play the "I'm not a native English speaker" card?


Not a problem, I appreciate being corrected. It came across much more as a gentle rebuke :) and I would definitely not be able to tell that you are not a native English speaker!

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The multiplication procedure is chosen precisely so that it coincides with composition of linear maps. Simply notice that multiplying an n x n real matrix on the right by a column vector produces a linear combination of the columns of the matrix whose coefficients are the elements of the column vector.

(This interpretation of multiplication of a matrix by a column vector is very useful and is forgotten too often by users of matrix algebra. Many basic facts follow directly from it!)

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Quote:
Original post by nilkn
The multiplication procedure is chosen precisely so that it coincides with composition of linear maps.


This is the answer I would have given.

You can think of a real number 'z' not "as a real number," but instead as a linear map 'Z' that takes other real numbers and returns real numbers. The map is defined by,

Z(x) = z x .

In math-speak, real numbers and linear maps are related by an isomorphism.

Matrix multiplication is the generalization to linear maps between vector spaces.

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Just want to say thanks to everyone for replying! The Help&Knowledge is greatly appreciated.

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This isn't a complete justification, but perhaps it will provide some insight.

One way to think of multiplication of things is 1) take every possible combination, then 2) simplify what you get. This basic 2-step conception of "multiplication" can be applied in many settings (numbers, sets, tensors, groups, etc), and the distinction between different types of multiplication lies in the rules you choose for simplification in the second step.

"Multiplying" things that are completely comparable (numbers):

3*2 = 1+1+1
+1+1+1 = 6

"Multiplying" things that are completely incomparable (variables or sets):

(1a + 3b)*(2x + 1y) = ax + bx + bx + bx
+ax + bx + bx + bx
+ay + by + by + by = 2ax + 1ay + 6bx + 3by

"Multiplying" matrices (tensors) is a situation where things are sort of comparable but not completely. A nice thing about matrices is that they can be written as a weighted sum of simple matrices with a 1 in the ij position and zeros everywhere else (ei ej^t), eg:

(1 2) = 1 e1 e1^t + 2 e1 e2^t + 3 e2 e1^t + 4 e2 e2^t = sum_ij(aij ei ej^t)
(3 4)

Taking every combination yields:

A*B = sum_ij(aij ei ej^t) * sum_kl(bkl ek el^t)
= sum_ijkl(aij bkl ei ej^t ek el^t)

And then the "simplification" rule for matrix multiplication is that (ej^t ek) = 0 unless j=k in which case it is 1. It is worth taking some time to think about why it is defined this way, but with the rule so chosen, the result simplifies to:

A*B = ... = sum_ikl(aik bkl ei el^t)

This is the same as saying (A*B)_il = sum_k(aik bkl), which you may recognize as the standard algorithm for computing the ij'th entry of a matrix product.

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Quote:
Original post by djz
I don't understand why the process of adding and summing is called 'multiplication', and not something else.
If I were asked to 'multiply' the two matrices, I would naturally come up with
[]. Is there a name for this?
I've seen 'matrix multiplication' also referred to as 'matrix concatenation' before, which isn't a bad name seeing as regular multiplication is commutative (whereas "matrix multiplication" and any kind of "concatenation" isn't).

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