Linear algebra and matrices M3P9 Question Sheet 2

Assessed coursework, solutions due 8.11.99.

1. Let

A = ( \begin{matrix}1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{matrix} )

Find a non-singular matrix P and a diagonal matrix B such that P^{−1}AP = B.

2. What matrices have linear minimal polynomial?

3. Find the minimal polynomials of the following matrices (λ and μ are in the ground field k):

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

A_{3} = ( \begin{matrix}λ & 1 & 0 & 0 \\ 0 & λ & 1 & 0 \\ 0 & 0 & λ & 0 \\ 0 & 0 & 0 & μ \end{matrix} ), A_{4} = ( \begin{matrix}λ & 1 & 0 & 0 \\ 0 & λ & 0 & 0 \\ 0 & 0 & μ & 0 \\ 0 & 0 & 0 & μ \end{matrix} ).

4. Suppose A is a square 4 × 4-matrix such that A^{4} = Id over the field of real numbers. Least all the possibilities for the minimal polynomial of A, and find matrices with such minimal polynomials.