It occurs to me, upon reviewing my own work that I've encountered a couple of new theorems in linear algebra. The first is this, which I know to be true, but have never bothered to prove.

GF(2) is the finite field which is also known as the integers modulo 2. We can build matrices from elements of GF(2) and manipulate these matrices just as we can matrices over the real or complex numbers.

In the following, I'm only interested in square matrices.

A matrix over GF(2) is in K form (my terminology) if it has ones on the main diagonal and zeros elsewere, except for one entry which is equal to 1.

The following matrices are in K form.

11000
01000
00100
00010
00001

1001
0100
0010
0001

Let M be an arbitrary matrix over GF(2). Then M can be expressed as:

Q x K1 x K2 x ... x Kn x Qi

where Q is an arbitrary non-singular matrix Qi is the inverse of Q, and K1 through Kn are matrices in K form.

Possible improvements:

1. Insist that Q is  a permutation matrix.
2. Eliminate Q entirely.

Another unproven theorem (which I'm certain is correct) is the following.

Suppose f=gM where g is a totally symmetric function and M is a non-singular matrix. Then
f=hN where h is a totally symmetric function and N is a product of K-form matrices.

Anybody have proofs?