MASTERS COMPREHENSIVE EXAM ADVANCED LINEAR ALGEBRA JANUARY 1999

(1) Consider the following matrices:

\begin{matrix}A_{1} = ( \begin{matrix}1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) & A_{2} = ( \begin{matrix}1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) & A_{3} = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) \\ A_{4} = ( \begin{matrix}1 & 1 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) & A_{5} = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ − − 1 & 0 & 0 & 1 \end{matrix} ) & A_{6} = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix} ) \end{matrix}

(a) Group these matrices into similarity classes. (Recall A_{j} is similar to A_{k} if and only if A_{j} = QA_{k}Q^{−1} for some invertible matrix Q. Then for each A_{j} group together all A_{k}’s similar to A_{j}.)

(b) Find the minimal polynomial of each A_{j}.

(2) What condition on the complex numbers α, β and γ is equivalent to the matrix

( \begin{matrix}0 & α & 0 \\ β & 0 & 0 \\ 0 & 0 & γ \end{matrix} )

being diagonalizable (as a complex matrix)?

(3) Consider the matrix

A = ( \begin{matrix}4 & 7 \\ −2 & −4 \end{matrix} ) .

(a) Find the eigenvalues of A.

(b) Find the eigenvalues of A^{15} + 2A + I. (Hint: You do not need to calculate powers of A.)

(4) (a) Is there a linear map T : ℝ^{4} → ℝ^{3} such that

T ((1, 3, 0, 1)) = (1, 0, 0),
T ((1, 1, 1, −1)) = (1, 1, 0), and
T ((−1, 3, −3, 5)) = (1, 0, 1)?

Explain your answer.

(b) Is there a linear map S : ℝ^{4} → ℝ^{3} such that

S((1, 2, 1, 1)) = (1, 2, 0),
S((1, 0, 1, 0)) = (1, 1, 0), and
S((0, 3, 0, 1)) = (1, 2, 1)?

Explain your answer.