Paper 1, Section II, A

Vectors and Matrices | Part IA, 2014

Let AA be a real n×nn \times n symmetric matrix.

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

(ii) Consider the linear system

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

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

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

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