Page "Symmetric matrix" Paragraph 14

The sum and difference of two symmetric matrices is again symmetric, but this is not always true for the product: given symmetric matrices A and B, then AB is symmetric if and only if A and B commute, i. e., if AB = BA.

So for integer n, A < sup > n </ sup > is symmetric if A is symmetric.

If A and B are n × n real symmetric matrices that commute, then there exists a basis of such that every element of the basis is an eigenvector for both A and B.