Paper 1, Section II, B

Vectors and Matrices | Part IA, 2016

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

The elements of MM are given by

Mij={0 for i=j1 for ijM_{i j}= \begin{cases}0 & \text { for } i=j \\ 1 & \text { for } i \neq j\end{cases}

Determine the elements of M2M^{2} and hence show that, if λ\lambda is an eigenvalue of MM, then

λ2=(n1)+(n2)λ\lambda^{2}=(n-1)+(n-2) \lambda

Assuming that MM can be diagonalised, give its diagonal form.

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