EECS 551 Midterm Exam, 10/12/2022, 6 – 8 pm
1. If A, B, C, D are matrices such that the product ABCD is defined (conformable), then
(ABCD)′ = D′ C′B′A′ . (2 pts)
True
False
2. If P is an orthogonal projection matrix, then ‖(I − P)x ‖2 ≤ ‖x ‖2 (2 pts)
True
False
3. If Y and Z are both unitary matrices and B = YAZ, then B and A have the same singular values, assuming the matrix sizes match. (2 pts)
True
False
4. If x ∈ ℝ3 andy ∈ ℝ4 are nonzero vectors, then the nullity of xy′=2 (2 pts)
True
False
5. Let b1, … , bN be an orthonormal set of vectors in FM and define matrix B = [b1, … , bN ].
Then ‖Bx‖2 = ‖x ‖2 for any x ∈ FN . (2 pts)
True
False
6. For A = [3 −4], ‖Ax‖2 is maximized for unit norm x when (2 pts)
True
False
7. If B is an orthogonal projection matrix and B is also an orthogonal matrix, then B = I. (2 pts)
True
False
8. If A+ = A′ then A has orthonormal columns. (2 pts)
True
False
9. If A is invertible, then AA+ = I (2 pts)
True
False
10. Let A be a Hermitian symmetric matrix and Q be a unitary matrix. If x is a unit-norm eigenvector of A, then Q)1x is a unit-norm eigenvector of B = Q′AQ. (2 pts)
True
False
11. If x andy are two non-zero vectors in ℂN, then ‖x + y‖2 = ‖x ‖2 + ‖y‖2 , if y = αx for any non-zero α ∈ ℝ . (2 pts)
True
False
12. Let b1, … , bN be a set of non-zero vectors in ℝN that are all pairwise perpendicular to each other. Then the matrix B = [b1, … , bN ] is an orthogonal matrix. (2 pts)
True
False
13. If A is an M × N matrix with rank N, where M ≥ N, then A+A = I. (2 pts)
True
False
14. If A is an M × N matrix, then R(A′) = R(A′A). (2 pts)
True
False
15. If A is an M × N matrix, then R(A) = R(AA+ ). (2 pts)
True
False
16. The matrix for which P2 = I, has all non-zero eigenvalues. (2 pts)
True
False
17. Define the matrix A ∈ R2N×2N such that for any x ∈ RN andy ∈ RN :
Then the matrix A is idempotent. (2 pts)
True
False
18. If B is a normal matrix and z is any unit-norm eigenvector of B, then there is an SVD of B where z is one of the left singular vectors. (2 pts)
True
False
19. If A is an M × N matrix with rank N, where M ≥ N, then minx ⅡAx — yⅡ2 = 0. (2 pts)
True
False
20. If C = [A B] then minx ⅡCx — yⅡ2 > minz ⅡAz — yⅡ2, assuming dimensions match appropriately. (2 pts)
True
False
21. If B is a 200 × 400 matrix with rank 100, then: (2 pts)
a. dim]R(B)^ = 100
b. dim]R丄 (B)^ = 100
c. dim]N(B)^ = 100
d. dim]N丄 (B)^ = 100
e. The number of distinct singular values is at least 2
22. When the unit-norm vector x that maximizes ‖Ax‖2 is: (2 pts)
a. [1 2]'/√5
b. [6 4 2]'/√14
c. [2 4 6]'/√14
d. [2 1]'/√5
e. [3 2 1]'/√14
f. None of the above
23. The value displayed by the JULIA code A=ones(2,8); norm(A,2) is (2 pts)
a. 2
b. 4
c. 8
d. 16
e. 32
24. Let u and v be two orthogonal vectors in F' . The number of non-zero eigenvalues of the matrix uv′ is (2 pts)
a. 0
b. 1
c. N-1
d. N
e. None of the above
25. For an M × N matrix with rank r, the number of singular values is (2 pts)
a. r
b. M
c. N
d. M + N
e. MN
f. min(N, M)
g. None of the above
26. Let V ∈ F'×' denote a unitary matrix. For y ∈ F', the most computationally efficient
JULIA code for solving argmin+∈F! ‖Vx − y‖! is: (2 pts)
a. pinv (V) * y
b. V \ y
c. V * y
d. V’ * y
e. Inv(V) * y
27. Let α and β denote the spectral norms of matrices A and B, respectively. The spectral norm of the matrix is (2 pts)
a. αβ
b. α + β
c. √α2 + β2
d. min (α, β)
e. max (α, β)
f. None of the above
28. Let A be an M × N matrix with non-zero rank. The orthogonal complement of the null space of A+ is. (2 pts)
a. R(A)
b. R(A9)
c. N(A)
d. N(A9)
e. R丄 (A)
f. R丄 (A′)
g. N丄 (A)
h. N丄 (A′)
29. Let A be a tall matrix having rank r with SVD given by
Define Pr丄 = I − urur ′ and P0(丄) = I − u0 u0 ′ and B = P0(丄)Pr丄 . Then: (2 pts)
a. B is a unitary matrix.
b. B is not a unitary matrix.
c. Need more information to assess
30. If B is an N × N idempotent matrix with trace{B} = K, then rank(B) is (2 pts)
a. 0 or 1
b. K
c. N − K
d. N
e. None of the above
31. The vectors {b1, b2, b3 } form. an orthonormal basis for a subspace S of ℝN, for N ≥ 5.
Complete the following JULIA function so that given input vector x ∈ ℝN, it returns the nearest vector in S. For full credit, your code must use as few floating-point calculations as possible. (4 pts)
function neareast (x, b1, b2, b3)
return
end
32. Determine the spectral norm of for b ∈ ℝ . (4 pts)
33. A simple linear system takes as input n (possibly complex) scalar values and returns as output the sum of those values. Thinking of the input as an n-dimensional vector, what unit norm input vector produces an output value with the largest possible magnitude? (4 pts)
34. Determine what will be displayed as output by the following JULIA code: (4 pts)
B = [3 0 0 -4];
pinv(B)
35. Complete the following JULIA function so that it returns the nullity of a matrix argument. (4 pts)
function nullity ( A )
(M,N)=size(A)
return
end
36. Determine the output value displayed by the following JULIA code. (4 pts)
A = 7 * ones(5,3);
(U, s, V) = svd(A);
B = U[:,1] * s[1] * V[:,1]’;
Vecnorm ( A – B )
37. Complete the following JULIA function so that it returns an orthonormal basis (as a matrix) for the span of the four input vectors a, b, c, d (assumed to be of the same length). (4 pts)
function spanbasis (a, b, c, d)
end
38. Let U and V denote unitary N × N matrices. Complete the following JULIA function so that
given input vector y ∈ ℂN, it returns the linear least-squares solution
argminx ‖UVx − y‖2(2).
For full credit, your code must use as few floating-point calculations as possible. (4 pts)
function best (y, U, V)
return
end
39. Determine a simple expression for the solution to the following regularized least-squares
cost function for δ > 0. x = argminxf(x) where f(x) = 2/1 ‖Ax − y‖2(2) + 2/1 δ 2 ‖Cx‖2(2), where C has full column rank. Your final expression should not have any pseudo-inverse in it and should be in terms of the original problem variables: A, C, y, δ . (8 pts)