Linear algebra and matrices M3P9

Question Sheet 1

In this sheet the ground field is the field of real numbers.

1. Which of the following pairs of matrices A, B are similar? Find a non-singular matrix P with P^{−1}AP = B if A and B are similar; otherwise, prove that no such matrix P exists.

(1) A = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) and B_{x} = ( \begin{matrix}0 & x \\ 0 & 0 \end{matrix} ), where x is a real number.

(2) A = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) and B = ( \begin{matrix}5 & 9 \\ 4 & 2 \end{matrix} ).

(3) A = ( \begin{matrix}1 & 1 \\ 0 & 2 \end{matrix} ) and B = ( \begin{matrix}1 & 0 \\ 0 & 2 \end{matrix} ).

(4) A = ( \begin{matrix}1 & 5 \\ 0 & 1 \end{matrix} ) and B = ( \begin{matrix}1 & 0 \\ 5 & 1 \end{matrix} ).

2. Find the characteristic polynomials of the following matrices (a and b are real numbers):

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

3. Find the (real) eigenvectors of the matrices in the previous exercise. Are any two of them similar?

4. Prove that every 3 × 3 (real) matrix has an eigenvector.

5. What matrices in Exercises 1 and 2 are diagonalizable (similar to a diagonal matrix)?