Exercise 10

A square matrix is symmetric if each $i,j$ entry equals the $j,i$ entry, that is, if the matrix equals its transpose.

• Prove that for any square $H$, the matrix $H+{{H}^{\mathsf{T}}}$ is symmetric. Does every symmetric matrix have this form?

• Prove that the set of $n\!\times\!n$ symmetric matrices is a subspace of $\mathcal{M}_{n\!\times\!n}$.