Exercise 18

A matrix is symmetric if for each pair of indices $i$ and $j$, the $i,j$ entry equals the $j,i$ entry. A matrix is antisymmetric if each $i,j$ entry is the negative of the $j,i$ entry.

• Give a symmetric $2\!\times\!2$ matrix and an antisymmetric $2\!\times\!2$ matrix. (Remark. For the second one, be careful about the entries on the diagonal.)

• What is the relationship between a square symmetric matrix and its transpose? Between a square antisymmetric matrix and its transpose?

• Show that $\mathcal{M}_{n\!\times\!n}$ is the direct sum of the space of symmetric matrices and the space of antisymmetric matrices.