Linear algebra and matrices M3P9

Question Sheet 5

1. Let

A = ( \begin{matrix}5 & 4 & 0 \\ −2 & −1 & 0 \\ 0 & 0 & 3 \end{matrix} ) , B = ( \begin{matrix}8 & 6 & 6 \\ −3 & −1 & −3 \\ −6 & −6 & −4 \end{matrix} )

Verify that AB = BA, and find an invertible matrix P such that P^{−1}AP and P^{−1}BP are both diagonal.

2. Let A = ( \begin{matrix}4 & 2 & −1 \\ 2 & 0 & −3 \\ −1 & −3 & 2 \end{matrix} ), and let < x, y >= x^{t}Ay be the bilinear

form whose Gram matrix is A. Find a basis of R^{3} with respect to which the Gram matrix of <, > is diagonal. Find a vector v such that < v, v >< 0.

3. Find the signature of the symmetric bilinear form whose Gram matrix is

( \begin{matrix}0 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 \end{matrix} )

4. Find the canonical form of the alternating bilinear form

( \begin{matrix}0 & −1 & 0 & 0 & −1 \\ 1 & 0 & −1 & 0 & 0 \\ 0 & 1 & 0 & −1 & 0 \\ 0 & 0 & 1 & 0 & −1 \\ 1 & 0 & 0 & 1 & 0 \end{matrix} )