Paper 1, Section II, B

What does it mean to say that a matrix can be diagonalised? Given that the $n \times n$ real matrix $M$ has $n$ eigenvectors satisfying $\mathbf{e}_{a} \cdot \mathbf{e}_{b}=\delta_{a b}$, explain how to obtain the diagonal form $\Lambda$ of $M$. Prove that $\Lambda$ is indeed diagonal. Obtain, with proof, an expression for the trace of $M$ in terms of its eigenvalues.

The elements of $M$ are given by

$M_{i j}= \begin{cases}0 & \text { for } i=j \\ 1 & \text { for } i \neq j\end{cases}$

Determine the elements of $M^{2}$ and hence show that, if $\lambda$ is an eigenvalue of $M$, then

$\lambda^{2}=(n-1)+(n-2) \lambda$

Assuming that $M$ can be diagonalised, give its diagonal form.

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