Question

For any permutation matrix P, show that every matrix can be expressed as the sum of a matrix that commutes with P and another matrix that anticommutes with P.

Answer 1

Solution:

A is an arbitrary matrix:

A = B + C

Where BP=PB and CP=-PC are the relations

P is a permutation matrix so it's its own inverse: \(PP=I\)

Now I show that these definitions of B satisfies this relation:

\[B=\frac{A+PAP}{2}\]

\[BP = \frac{(A+PAP)P}{2} = \frac{AP+PA}{2} = \frac{P(A+PAP)}{2} = PB\]

The rest is pretty straight forward to show on your own, but feel free to ask questions if you have 'em!

Answer 2

This is what I do when I'm bored lol