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# 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 = 04x -  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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multiplying two matrices times a vector
(A*B)v = is the same thing as if you did
(A*B)v = A*B*v

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Quote:
 Original post by djzIs 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 alvarodavetyler'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 davetylerNot 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?