Linear algebra and matrices M3P9

Question Sheet 6

Assessed coursework, solutions due 13.12.99.

1. Prove the following statements or give a counterexample:

(1) If A and B are symmetric, then so is AB.

(2) If A and B are skew-symmetric, then so is AB.

(3) If A and B are skew-symmetric, then so is AB − BA.

2. For the following matrices A find an orthogonal matrix P such that P^{t}AP is diagonal:

A_{1} = ( \begin{matrix}2 & 1 \\ 1 & 4 \end{matrix} ) , A_{2} = ( \begin{matrix}0 & 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 & 0 \end{matrix} )

3. Prove that all eigenvalues of a skew-symmetric matrix with real coefficients are purely imaginary (= of the form λ\sqrt{−1}, λ ∈ R). Hint: imitate the argument for symmetric matrices from the lectures.

4. Let α and β be elements of the ground field k. Consider the vector space V of square matrices of size n with coefficients in k. If A and B are in V, define < A, B >= αtr(AB)+βtr(A)tr(B). Show that this is a symmetric bilinear form, and find its rank (for all α and β).