MASTER’S COMPREHENSIVE EXAMS Advanced Linear Algebra January 2000

1. Consider the matrix

A = [ \begin{matrix}3 & 4 \\ 4 & −3 \end{matrix} ] .

(a) Find the eigenvalues of A.

(b) Find a matrix Q so that Q^{−1} A Q is a diagonal matrix.

(c) Find the minimal polynomial of A.

(d) Find A^{n} for every positive integer n.

(e) Give a geometric description of A.

2. Let V be an inner product space i.e. a real vector space with an inner product which we denote by (x, y). Recall that a linear transformation T of V is said to be symmetric or self adjoint if (T (x), y) = (x, T (y)) for all x, y ∈ V. Let T be a symmetric linear map. Prove the following two facts.

(a) ker(T) and ran(T) are orthogonal.

(b) V = ker(T) ⊕ ran(T).

3. Let R^{n×n} be the vector space of all n × n matrices with real entries. For each nilpotent matrix A ∈ R^{n×n} consider the dimension of span{A^{k} : k = 0, 1, 2, …} (a subspace of R^{n×n}). What is the greatest this dimension can be? Explain your answer. (Here A^{0} is the identity matrix.)

4. Let p(x) = (x − 2)^{2}(x − 4)^{3}. Determine all minimal polynomials of 5 × 5 matrices that have p(x) as characteristic polynomial. Also find all Jordan canonical forms of such matrices.