Why are matrix and scalar multiplication denoted the same way and treated as the same operation in standard mathematical notation? This is always a source of confusion for me because they have completely different properties (specifically commutativity). Multiplying a 1x1 matrix by an NxN matrix isn't even generally equivalent to multiplying an NxN matrix by a scalar. (The former is not even always defined.) Wouldn't it be clearer to consider these to be completely unrelated operations and use completely different notation to represent them?
What does matrix multiplication have to do with scalar multiplication?
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0They are not "treated as the same operation"; we call them both "multiplication", but that's not the same as "treat[ing them] as the same operation". We call addition of vectors "sum", just like sum of numbers, even when it has nothing to do with sums of numbers (the set of positive reals is a vector space over the reals with vector addition $\mathbf{u}\oplus\mathbf{v}=uv$ and scalar product $\alpha\odot\mathbf{v}=v^{\alpha}$). We call them "multiplication" because they have enough similar properties to warrant it. – 2011-03-29
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1That said, "multiplication" of matrices is best thought of as and analogue of "composition" (just like composition of functions). It's just that the set of $N\times N$ matrices, for a fixed $N$, form a ring http://en.wikipedia.org/wiki/Ring_%28mathematics%29 under addition of matrices and matrix composition/multiplication, and the second operation of rings is always called "multiplication". – 2011-03-29
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0@Arturo: Good point, but vector addition maps to scalar addition in a much more transparent way (it's just the component-wise sum) and preserves important properties like commutativity. It's subjective, but I think scalar vs. matrix multiplication stretches things much further than scalar vs. vector addition. Thus from a notational clarity point of view, denoting vector and scalar addition the same way is more reasonable than denoting scalar and matrix multiplication the same way. – 2011-03-29
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0No, vector addition is *not* always "component-wise sum". That's the point of my example above. For what it may be worth, "scalar multiplication" is definitely *different* from "multiplication". Multiplication is a binary operation on a set (takes two arguments from the set, returns an element of the set), whereas "scalar multiplication" is not an operation under that definition. In General Algebra, "scalar multiplication" is viewed as an entire family of unary operations instead. – 2011-03-29
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0By "scalar multiplication" I understood the vector space map that takes a scalar and a vector and returns a vector; if you meant "multiplication of 'scalars'/'numbers'", then my last comment probably reads like nonsense. – 2011-03-29