Paper 1, Section II, B

Vectors and Matrices | Part IA, 2011

(a) Let MM be a real symmetric n×nn \times n matrix. Prove the following.

(i) Each eigenvalue of MM is real.

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

(iii) Eigenvectors with different eigenvalues are orthogonal.

(b) Let AA be a real antisymmetric n×nn \times n matrix. Prove that each eigenvalue of A2A^{2} is real and is less than or equal to zero.

If λ2-\lambda^{2} and μ2-\mu^{2} are distinct, non-zero eigenvalues of A2A^{2}, show that there exist orthonormal vectors u,u,w,w\mathbf{u}, \mathbf{u}^{\prime}, \mathbf{w}, \mathbf{w}^{\prime} with

Au=λu,Aw=μwAu=λu,Aw=μw\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

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