# 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

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 $$\displaystyle i \geq 2$$