Discomposition of a matrix

[KLHON]

Let C be a squared matrix of order n, also the sum of the diagonal entries, i.e. trace of C, is zero. Is there exist two squared matrices of order n such that C=AB-BA. I am most interested in the proceedure of getting A and B

Thank for your help

Sincerely Yours

Edmond

 

[Idea]

For \(\displaystyle i \neq j\), the elementary matrices \(\displaystyle E_{i j}\) , \(\displaystyle (1\) in the \(\displaystyle i,j \) position and \(\displaystyle 0\) everywhere else)

can be written in the required form since

\(\displaystyle E_{i j}=E_{i 1}E_{1j}-E_{1 j}E_{i 1}\)

Any diagonal matrix whose trace is \(\displaystyle 0\) can be written as a linear combination of matrices of the form

\(\displaystyle E_{1 1}-E_{i i}\) where \(\displaystyle i \geq 2\)