# When can a matrix be represented, up to a permutation, as symmetric or diagonal?

by David J. Harris   Last Updated October 02, 2017 15:20 PM

Suppose I have a square matrix, \$\textbf{A}\$. I want to know whether I can represent it, it up to a permutation of the columns, using a symmetric or triangular matrix instead, since these only have \${n}\choose{2}\$ entries instead of \$n^2\$.

In other words, what properties would \$\textbf{A}\$ need to have for a solution to exist for the following equation?

\$\$\textbf{A P} = \textbf{B},\$\$

where \$\textbf{P}\$ is a permutation matrix and \$\textbf{B}\$ is either triangular or symmetric (whichever is easier to write an answer about).

I don't need to actually find \$\textbf{B}\$ or compare individual pairs of \$\textbf{A}\$ and \$\textbf{B}\$, so this may be simpler than some previous questions such as this one or this one.

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