Exercise 6.8.12

Prove the three conditions.

reflexivity

We have $A$ is congruent to $A$ since $A=I^t\mathit{AI}$.

symmetry

If $A$ is congruent to $B$, we have $B=Q^t\mathit{AQ}$ for some invertible matrix $Q$. Hence we know that $B$ is congruent to $A$ since $A={(Q^{-1})}^{t}A{Q}^{-1}$.

transitivity

If $A$ is congruent to $B$ and $B$ is congruent to $C$, we have $B=Q^t\mathit{AQ}$ and $C=P^t\mathit{BP}$. Thus we know that $A$ is congruent to $C$ since $C={(\mathit{QP})}^{t}A(\mathit{QP})$.