A symmetric group is then a subgroup of the general linear group in any field, where the general linear group GL(K, N) is the set of invertible NxN matrices with entries from a "field" (just think "numbers") and therefore any [finitely generated] group is isomorphic to a group of matrices under matrix multiplication; this is the underlying concept of representation theory.
If we back up a little, a group is a very general sort of algebraic structure; many important concepts have an underlying group structure, such as rotations, permutations, and any sort of reversible computation. This latter case implies that matrix multiplication is Turing complete; the simplest such set of matrices is generated by the Toffoli gate matrix. The relationship of groups to geometry is due to the underlying correspondence between the axioms of a group and those of geometrical transformations. A set of reversible geometric transformations includes an identity element -- do nothing -- and obeys associativity (sorta complicated, but it makes sense if you think about it) and inverse operations (by assumption): this makes it a group, and it can be represented by matrices. If we remove "reversible", we get exceptions -- like the cross product -- but these are usually related to groups (cross product -> quaternion algebra).
So matrix multiplication is actually really, really fundamental in a lot of mathematics. It's also a special case of tensor contraction, which could justify another tower post (but won't).
>And trust me, when you are writing numerical code, being able to read the mathematical formula clearly is essential, especially when you have a lot of formulas and need to figure out why your code is giving you numerical non-sense.
(spent three months chasing a bug where two commands were out of order)
> A symmetric group is then a subgroup of the general linear group in any field, where the general linear group GL(K, N) is the set of invertible NxN matrices with entries from a "field" (just think "numbers") and therefore any [finitely generated] group is isomorphic to a group of matrices under matrix multiplication; this is the underlying concept of representation theory.
As a representation theorist, I am very sympathetic to this point of view, but I'm not sure that it proves that "[almost] every single linear algebra routine can be interpreted as a matrix-matrix multiplication"—unless one first has some reduction from an arbitrary linear-algebra routine to a group action.
> A symmetric group is then a subgroup of the general linear group in any field, where the general linear group GL(K, N) is the set of invertible NxN matrices with entries from a "field" (just think "numbers") and therefore any [finitely generated] group is isomorphic to a group of matrices under matrix multiplication; this is the underlying concept of representation theory.
Also, as a very minor nitpick, I think that you want 'finite' instead of 'finitely generated'; even infinite groups embed in (infinite) symmetric groups, but it's not obvious to me that infinite symmetric groups embed in (finite-dimensional) matrix groups, and it's certainly not true (just by counting cardinality) that infinite but finitely generated groups embed in finite symmetric groups.
It's not really necessary to argue. Cayley's theorem guarantees that every group is a subgroup of a symmetric group:
http://en.wikipedia.org/wiki/Cayley's_theorem
A symmetric group is then a subgroup of the general linear group in any field, where the general linear group GL(K, N) is the set of invertible NxN matrices with entries from a "field" (just think "numbers") and therefore any [finitely generated] group is isomorphic to a group of matrices under matrix multiplication; this is the underlying concept of representation theory.
If we back up a little, a group is a very general sort of algebraic structure; many important concepts have an underlying group structure, such as rotations, permutations, and any sort of reversible computation. This latter case implies that matrix multiplication is Turing complete; the simplest such set of matrices is generated by the Toffoli gate matrix. The relationship of groups to geometry is due to the underlying correspondence between the axioms of a group and those of geometrical transformations. A set of reversible geometric transformations includes an identity element -- do nothing -- and obeys associativity (sorta complicated, but it makes sense if you think about it) and inverse operations (by assumption): this makes it a group, and it can be represented by matrices. If we remove "reversible", we get exceptions -- like the cross product -- but these are usually related to groups (cross product -> quaternion algebra).
So matrix multiplication is actually really, really fundamental in a lot of mathematics. It's also a special case of tensor contraction, which could justify another tower post (but won't).
>And trust me, when you are writing numerical code, being able to read the mathematical formula clearly is essential, especially when you have a lot of formulas and need to figure out why your code is giving you numerical non-sense.
(spent three months chasing a bug where two commands were out of order)