Let $A$ be a real symmetric $n \times n$ matrix. Prove that every eigenvalue of $A$ is real, and that eigenvectors corresponding to distinct eigenvalues are orthogonal. Indicate clearly where in your argument you have used the fact that $A$ is real.

What does it mean to say that a real $n \times n$ matrix $P$ is orthogonal ? Show that if $P$ is orthogonal and $A$ is as above then $P^{-1} A P$ is symmetric. If $P$ is any real invertible matrix, must $P^{-1} A P$ be symmetric? Justify your answer.

Give, with justification, real $2 \times 2$ matrices $B, C, D, E$ with the following properties:

(i) $B$ has no real eigenvalues;

(ii) $C$ is not diagonalisable over $\mathbb{C}$;

(iii) $D$ is diagonalisable over $\mathbb{C}$, but not over $\mathbb{R}$;

(iv) $E$ is diagonalisable over $\mathbb{R}$, but does not have an orthonormal basis of eigenvectors.