Math 316
Carter
Test 2 Spring 2011
General Instructions: Do not write on this test. Write your name on only the outside
of your blue books. Do all your work inside your blue books. On the inside of your blue
book, write neat complete answers to the questions below. Show your work. This test has
nothing to do with Keanu Reeves. Good luck.
1. (20 points) Consider the ordered orthonormal basis
√
√
√
√
B = {Q1 , Q2 } = {[1/ 5, 2/ 5]t , [−2/ 5, 1/ 5]t }.
Let x0 , y 0 be the B coordinates of a point in the plane. Write the equation of the ellipse
25(x0 )2 + 4(y 0 )2 = 100
in standard form.
2. (10 points) Let A ∈ M (2, 2) denote a symmetric (2 × 2)-matrix (so A = At ). A
consequence of the spectral theorem is that
QD = AQ
where Q is a (2 × 2) orthogonal matrix and D is a diagonal matrix whose non-zero
entries are the (real)-eigenvalues of A. Describe this result in terms of a square, a
rectangle, a rotation, and possibly a reflection. For convenience assume that A is
non-singular. How can you avoid incorporating a reflection?
3. (5 points) Let A ∈ M (n, n) denote a symmetric matrix. Prove that for all vectors
X, Y ∈ M (n, 1),
(AX) · Y = X · (AY ).
4. (10 points) Use the result of the previous problem to show that if X and Y are eigenvectors for a symmetric matrix A with distinct eigenvalues, then X · Y = 0. [Hint: Let
AX = λX, and consider (λX) · Y ].
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#
a b
5. (20 points) Let A =
denote a (2 × 2) symmetric matrix. By directly comb d
puting roots of the characteristic polynomial (quadratic formula) for A, show that the
eigenvalues of A are real. [Hint: The sum of two perfect squares is positive].
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6. Let A ∈ M (m, n) denote an (m × n)-matrix with rank(A) = r.
(a) (5 points) Prove that B = At A is symmetric.
(b) (10 points) Prove that rank(B) = r.
(c) (10 points) Prove that the eigenvalues of B are positive real numbers. (You may
assume the spectral theorem and that B = U EU t for E ∈ M [r, r] and U a
partially orthogonal (n × r)-matrix.
7. (10 points) Give a complete statement of either the reduced singular value theorem or
the spectral theorem.
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