MASTER’S COMPREHENSIVE EXAMS Advanced Linear Algebra Three problems completed correctly out of four will constitute a passing score on this exam.

1. Let V be a complex vector space.

(a) Define what it means for ⟨·, ·⟩ : V × V → C to be an inner product on V.

(b) Prove the polar identity, that

4⟨u, v⟩ = ||u + v||^{2} − ||u − v||^{2} + i||u + iv||^{2} − i||u − iv||^{2}

for all u, v ∈ V.

2. Recall that, for an n × n matrix A = (a_{ij}), we define the trace of A by Tr(A) = ∑_{j=1}^{n} a_{jj}.

(a) Prove that Tr is linear.

(b) Prove that, for any two n × n matrices A and B, Tr(AB) = Tr(BA).

(c) Prove that there do not exist n × n matrices A and B such that AB − BA = I_{n}, where I_{n} is the n × n identity matrix.

3. Let A be the matrix

A = [ \begin{matrix}7 & −1 & 3 \\ 0 & 4 & 0 \\ −5 & 7 & −1 \end{matrix} ] .

(a) Find the characteristic polynomial of A and the eigenvalues of A.

(b) Find the eigenvectors for A. Is A diagonalizable?

(c) Give the Jordan form J of A.

(d) Find an appropriate basis for R^{3} with respect to which the matrix for A is in its Jordan form J, and give the change of basis matrix P so that J = P^{−1}AP. (Note: you are not required to find the matrix P^{−1}.)

4. Let A and B be two n × n diagonalizable real matrices.

(a) If A and B have the same characteristic polynomial, prove that they are similar.

(b) If A and B have the same minimal polynomial, need they be similar? Either prove this statement or give a counterexample.