Paper 1, Section II, A

Let $A$ be a real $n \times n$ symmetric matrix.

(i) Show that all eigenvalues of $A$ are real, and that the eigenvectors of $A$ with respect to different eigenvalues are orthogonal. Assuming that any real symmetric matrix can be diagonalised, show that there exists an orthonormal basis $\left\{\mathbf{y}_{i}\right\}$ of eigenvectors of $A$.

(ii) Consider the linear system

$A \mathbf{x}=\mathbf{b} .$

Show that this system has a solution if and only if $\mathbf{b} \cdot \mathbf{h}=0$ for every vector $\mathbf{h}$ in the kernel of $A$. Let $\mathbf{x}$ be such a solution. Given an eigenvector of $A$ with non-zero eigenvalue, determine the component of $x$ in the direction of this eigenvector. Use this result to find the general solution of the linear system, in the form

$\mathbf{x}=\sum_{i=1}^{n} \alpha_{i} \mathbf{y}_{i}$

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