Math 510 Final Examination 12 December 2006
Directions: All answers must be justified by computation or explanation. Greater weight will be given to one
whole (correct) solution than to two error-free but incomplete solutions. Six complete correct answers will receive
full credit, but you may answer additional questions if desired. Write each solution on a separate page. Submit
solutions in the same order as the questions.
1. The characteristic polynomial of A∈Q11×11 is p(x) = (x−1)5(x2+ 1)3, the minimum polynomial
of Ais p(x) = (x−1)3(x2+ 1)2, and the geometric multiplicity of eigenvalue λ= 1 is 3. Determine
the invariant factors of A.
2. Let Abe a normal matrix. Prove that A∗commutes with any matrix that commutes with A.
3. (a) Assume A∈Fn×nis idempotent. Prove that the rank of Ais equal to the number of nonzero
eigenvalues of A.
(b) Give a counterexample to show that the statement in 3a may be false without the assumption
that Ais idempotent.
4. Find the reduced singular value decomposition (i.e., find singular values and right and left singular
vectors for nonzero singular values) for A=
1 2
2 0
0 2
.
5. Let A, B ∈Cn×nbe Hermitian. Prove that λmax(A−B) + λmin(B)≤λmax (A).
6. Let A∈Cn×nand λ∈C. Prove that
rank((A−λI)2) + rank((A−λI )8)≥rank((A−λI )5) + rank((A−λI)5).
7. Let A∈Fn×nand let λ∈σ(A). Prove that Ais similar to a matrix having the sum of the elements
in each row equal to λ.
8. Let H∈Cn×nbe Hermitian and partition Has H=H11 H12
H21 H22with H11 , H22 square and H11
nonsingular. Prove that the Schur complement H/H11 is Hermitian.
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