Practice Qualifying Exam B [pdf]

Masters Comprehensive Exam in
Matrix Analysis (Math 603)
Practice Exam B, August 2010
Do any three problems. Show all your work. Each problem is worth 10 points.
(1) Suppose that A is a 3 × 3 matrix with eigenvalues λ 1 = 2 and λ 2 = 3 and eigenspaces
⎧⎛
⎞ ⎛ ⎞⎫
⎧⎛
⎞⎫
⎨ 1 0 ⎬
⎨ 1 ⎬
N (A − 2I) = Span ⎝0⎠ , ⎝1⎠
N (A − 3I) = Span ⎝ 0 ⎠
⎩
⎭
⎩ ⎭ .
1 0
−1
(a) Show that the function f : R 3 → R defined by f(x) = x T Ax is positive for all x ≠ 0.
(b) Calculate the spectral projectors G 1 and G 2 corresponding to λ 1 and λ 2 .
(c) Express A in terms of G 1 and G 2 .
(2) (a) Let A be an invertible n × n matrix with complex entries. Prove that
〈x|y〉 A := x ∗ A ∗ Ay
is an inner product on C n .
(b) Let {v 1 , v 2 , · · · , v n } be an orthonormal basis for a complex inner-product space (V, 〈·|·〉).
(i) Prove that
n∑
〈x|y〉 = 〈x|v i 〉〈v i |y〉 for all x and y in V.
∑
(ii) Suppose that x = n ∑
λ i v i . Prove that |x| 2 = n |λ i | 2 .
i=1
i=1
i=1
(3) (a) Prove that similar matrices have the same spectrum.
(b) Use the result of (a) to define the spectrum of a linear transformation T : V → V and prove
that it is well defined. (Here V is a finite dimensional vector space.)

(4) (a) Let A be an n × n matrix. A generalized eigenvector of A is a nonzero vector v so that
(A − λI) k v = 0 for some integer k > 0 and λ ∈ C. Construct a matrix A and a vector v such that
v is a generalized eigenvector of A but not an eigenvector of A.
(b) Write down all possible 3 × 3 Jordan matrices that have eigenvalues 2 and 5 (and no others).
(5) (a) Prove that the set of all n × n real skew-symmetric matrices is a vector space. Find a basis
for this vector space, and prove that it is indeed a basis.
(b) Prove that the eigenvalues of a real skew-symmetric matrix must be pure imaginary.