Linear algebra and matrices M3P9

Question Sheet 4

Assessed coursework, solutions due 22.11.01.

1. Let J_{λ,n} be the Jordan block of size n with λ. If f(t) is a polynomial, show that

f(J_{λ,n}) = ( \begin{matrix}f(λ) & f′(λ)/1! & f′′(λ)/2! & … & f^{(n−1)}(λ)/(n − 1)! \\ 0 & f(λ) & f′(λ)/1! & … & f^{(n−2)}(λ)/(n − 2)! \\ … & & & & … \\ 0 & 0 & 0 & … & f(λ) \end{matrix} )

Here f′(t), f′′(t), ..., f^{(n−1)}(t) are derivatives.

2. (1) Let A be an upper triangular matrix (a_{ij} = 0 if i > j) such that a_{ii} = c, i = 1, …, n, for some c ∈ k (all entries in the main diagonal are equal). When A is diagonalizable?

(2) Let λ ≠ μ and a, b, c be in the ground field k. When is the matrix

( \begin{matrix}λ & a & b \\ 0 & λ & c \\ 0 & 0 & μ \end{matrix} )

diagonalizable?

3. Find the Jordan normal form of the following matrices:

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

4. Find the minimal possible n for which there exist two non-similar matrices A and B of size n such that

(a) f_{A}(t) = f_{B}(t),

(b) m_{A}(t) = m_{B}(t),

(c) f_{A}(t) = f_{B}(t) and m_{A}(t) = m_{B}(t).