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Is matrix multiplication equivalent to numeric multiplication, or do they just share the same name?

While there are similarities between how they work, and one can be thought of being derived from the other, I ask because they have different properties such as not being commutative, a × b ≠ b × a, and sometimes multiplication between matrices is referred to by the alternative name apply instead of multiply. For example applying a transformation matrix, where this is the same as multiplying by it.

Additionally sometimes in programming operations can be defined between new types of things, allowing the language to expand with new concepts, however the link between the name and rules such as commutative are supposed to continue to hold true.

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"Is matrix multiplication equivalent to numeric multiplication" Equivalent in what sense? – Pete L. Clark Feb 10 '12 at 14:50
Matrix multiplication is associative. What it isn't is commutative, except in special cases like diagonal matrices... – J. M. Feb 10 '12 at 14:54
Thanks, I used the wrong word. – alan2here Feb 10 '12 at 14:57
You could, of course, think of 'numeric multiplication' as multiplying 1x1 matrices. – Sid Raval Feb 10 '12 at 15:14
up vote 4 down vote accepted

In every ring we use the name "multiplication" or "product" for the binary operation usually denoted $"."$. This multiplication is associative but it can be commutative or not.

On the other hand "apply" is not a synonymous of "multiply". You apply a transformation to an element which in the case of linear transformation is like multiply the vector by the associated matrix.

If you read a little about groups, rings and vector spaces you will understand better!

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Thanks. I'll look into it. – alan2here Feb 10 '12 at 15:00

"Multiplication" is often used to describe binary operations that share only some of the properties of ordinary multiplication.

The case of matrix multiplication is special. There, multiplication is in some sense a generalization of ordinary multiplication. Let $M_n(a)$ be the $n\times n$ matrix whose diagonal entries are all equal to $a$, and whose off-diagonal entries are $0$. It is easy to verify that $$M_n(x)M_n(y)=M_n(xy).$$ So we can think of the real numbers as, for example, special $7\times 7$ matrices. Then the multiplication of real numbers can be viewed as a special case of matrix multiplication.

More interestingly, define the $2\times 2$ matrix $M(a,b)$ by $$M(a,b)=\pmatrix{x & -y\\ y & x}.$$ It is not hard to verify that $$\pmatrix{a & -b\\ b & a}\pmatrix{c & -d\\ d & c}=\pmatrix{ac-bd & -(ad+bc)\\ ad+bc & ac-bd}.$$

Note that the product of the two complex numbers $a+ib$ and $c+id$ is equal to $ad+bc +i(ad+bc)$. So the special matrices $M(x,y)$ multiply like the complex numbers. They also add like the complex numbers, and can be identified with the complex numbers.

So the multiplication of $2\times 2$ matrices can be thought of as a generalization of the multiplication of complex numbers. This is of practical usefulness: rotations about the origin can be thought of as either multiplying by a special kind of complex number, or as multiplying by a special type of matrix.

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