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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?

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By proportional, do you mean related by a positive scalar?

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.

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  • $\begingroup$ Yes, I meant related by a positive scalar. So the conditions that I wrote could work if I chose the right norm? What about the Hilbert-Schmidt norm? $\endgroup$ – Ziofil Aug 29 '16 at 17:18

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