Paper 1, Section II, B

Vectors and Matrices | Part IA, 2017

(a) Show that the eigenvalues of any real n×nn \times n square matrix AA are the same as the eigenvalues of ATA^{T}.

The eigenvalues of AA are λ1,λ2,,λn\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n} and the eigenvalues of ATAA^{T} A are μ1,μ2,\mu_{1}, \mu_{2}, \ldots, μn\mu_{n}. Determine, by means of a proof or a counterexample, whether the following are necessary valid: (i) i=1nμi=i=1nλi2\sum_{i=1}^{n} \mu_{i}=\sum_{i=1}^{n} \lambda_{i}^{2}; (ii) i=1nμi=i=1nλi2\prod_{i=1}^{n} \mu_{i}=\prod_{i=1}^{n} \lambda_{i}^{2}.

(b) The 3×33 \times 3 matrix BB is given by

B=I+mnTB=I+\mathbf{m n}^{T}

where m\mathbf{m} and n\mathbf{n} are orthogonal real unit vectors and II is the 3×33 \times 3 identity matrix.

(i) Show that m×n\mathbf{m} \times \mathbf{n} is an eigenvector of BB, and write down a linearly independent eigenvector. Find the eigenvalues of BB and determine whether BB is diagonalisable.

(ii) Find the eigenvectors and eigenvalues of BTBB^{T} B.

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