# Paper 1, Section II, B

(a) Show that a square matrix $A$ is anti-symmetric if and only if $\mathbf{x}^{T} A \mathbf{x}=0$ for every vector $\mathbf{x}$.

(b) Let $A$ be a real anti-symmetric $n \times n$ matrix. Show that the eigenvalues of $A$ are imaginary or zero, and that the eigenvectors corresponding to distinct eigenvalues are orthogonal (in the sense that $\mathbf{x}^{\dagger} \mathbf{y}=0$, where the dagger denotes the hermitian conjugate).

(c) Let $A$ be a non-zero real $3 \times 3$ anti-symmetric matrix. Show that there is a real non-zero vector a such that $A \mathbf{a}=\mathbf{0}$.

Now let $\mathbf{b}$ be a real vector orthogonal to $\mathbf{a}$. Show that $A^{2} \mathbf{b}=-\theta^{2} \mathbf{b}$ for some real number $\theta$.

The matrix $e^{A}$ is defined by the exponential series $I+A+\frac{1}{2 !} A^{2}+\cdots$ Express $e^{A} \mathbf{a}$ and $e^{A} \mathbf{b}$ in terms of $\mathbf{a}, \mathbf{b}, A \mathbf{b}$ and $\theta$.

[You are not required to consider issues of convergence.]