(a) Let $M$ be a real symmetric $n \times n$ matrix. Prove the following.

(i) Each eigenvalue of $M$ is real.

(ii) Each eigenvector can be chosen to be real.

(iii) Eigenvectors with different eigenvalues are orthogonal.

(b) Let $A$ be a real antisymmetric $n \times n$ matrix. Prove that each eigenvalue of $A^{2}$ is real and is less than or equal to zero.

If $-\lambda^{2}$ and $-\mu^{2}$ are distinct, non-zero eigenvalues of $A^{2}$, show that there exist orthonormal vectors $\mathbf{u}, \mathbf{u}^{\prime}, \mathbf{w}, \mathbf{w}^{\prime}$ with

$$\begin{array}{rlr} A \mathbf{u}=\lambda \mathbf{u}^{\prime}, & A \mathbf{w}=\mu \mathbf{w}^{\prime} \\ A \mathbf{u}^{\prime}=-\lambda \mathbf{u}, & A \mathbf{w}^{\prime}=-\mu \mathbf{w} \end{array}$$

Part IA, 2011 List of Questions