# Algebraic condition for matrix proportionality?

I need a way of showing if two matrices are proportional to each other. I thought of two possible solutions: if I think of the two matrices as vectors, the triangle and the Cauchy-Schwarz inequalities are saturated. So the conditions would read $||A+B|| = ||A|| + ||B||$ or $||AB|| = ||A||\cdot||B||$. Are these okay? Are there better ways?

Depending on the norm you choose for the matrices, there are some problems with your conditions... Using the operator norm with respect to Euclidean distance your condition does not work. For example $A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ \end{pmatrix}$ satisfy $\|A+B\| = \|A\| + \|B\|$ and $\|AB\| = \|A\|\|B\|$, but aren't proportional.