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EECS 551 Midterm 1

1. Let x ∈ F N and y ∈ FM and U ∈ FM×N . Select the true statements.

a: If M = N and U is unitary then ∥Ux∥2 = ∥x∥2

b: If U is semi-unitary then ∥Ux∥2 = ∥x∥2

c: If M = N and U is unitary then ∥Ux − y∥2 = ∥x − U′y∥2

d: If U is semi-unitary then ∥Ux − y∥2 = ∥x − U′y∥2

e: None of these

2. If A and B are Hermitian symmetric matrices of the same size, then which of the following must be Hermitian matrices?

a: A + B

b: AB

c: AB′

d: αA for α ∈ F

e:

f: A+

g: None of these

3. Let B denote a Hermitian symmetric matrix. Which statements are true?

a: The eigenvalues of B are real.

b: The eigenvalues of B are non-zero.

c: Any singular value of B is also an eigenvalue of B.

d: Any right singular vector of B is also an eigenvector of B.

e: None of these

4. If B ∈ F N×N then which of the following statements are correct?

a: R(B) = F N

b: R(B) ⊆ F N

c: rank(B) = N

d: dim(R(B)) ≤ N

e: dim(R(B)) + dim(N (B)) ≤ N

f: dim(R(B)) > 0

g: dim(N (B)) > 0

h: None of these

5. Suppose matrix A has an SVD and y ∈ R(U2). If xˆ denotes the minimum-norm least-squares solution to Ax ≈ y, then which statements are true?

a: ∥xˆ∥ = 0

b: y ∈ R(A)

c: xˆ ∈ N (A)

d: None of these

6. Let X ∈ FM×N have rank 1 ≤ r ≤ min(M, N), and define Y = XX+XX′X. Let A denote the set of right singular vectors of X and let B denote the set of right singular vectors of Y . Which of these statements are true?

a: A = B

b: A is a strict subset of B

c: B is a strict subset of A

d: A has cardinality one

e: A is a subspace of dimension one

f: A is an empty set

g: B is an empty set

h: None of these

7. The projection of the vector onto the subspace span Determine z.

a: -1

b: 0

c: 1

d: 2

e: 3

f: 4

8. Matrix C ∈ F N×N has rank 0 < r < N and an SVD of C is C = [U1 U2] Σ [V1 V2] ′ , where U1 = [u1, . . . ,ur], V1 = [v1, . . . , vr] and Σ is diagonal.

Which of these is/are a projection matrix onto the null space of the Hermitian transpose of C:

a: I − U1U1 ′

b: I − V1V1 ′

c: U1U1 ′

d: V1V1 ′

e: V2U2 ′

f: V1U1 ′

g: U1V2 ′

h: U2V1 ′

9. Matrix A is N ×3 and matrix B is M ×3 and they each have an SVD that uses the same 3×3 right singular vector matrix V . A has singular values {1, 1, 2} and B has singular values {1, 2, 3}. Determine det{A′A + B′B} .

10. Hermitian matrix B has unitary eigendecomposition where V1, V2, and V3 have 4, 5 and 6 columns, respectively. Find the simplest expression for the projection matrix for N (B).

11. Hermitian matrix A has eigenvalues {−4, −1, 6}. Define B ≜ −5I + 3A. Determine the spectral norm of B.

12. Let A ∈ FM×N have rank r > 0 and have SVD A = UΣV ′ . Let Q denote a M ×M unitary matrix and define B ≜ QA. Specify an orthonormal basis for R⊥(B) in terms of (parts of) Q, U, Σ and/or V .

13. Matrix A has singular values 2/k for k = 1, . . . , 8 and 0 for k = 9, . . . , 20, with corresponding left singular vectors u1, . . . , u20. Determine ∥A+u6∥2 .

14. Let V = R 2×2 over the field R, the vector space of real 2×2 matrices. Let S denote the subspace of V consisting of Hermitian matrices. Specify the dimension of S and give a basis for S.

15. Given a matrix B ∈ CM×N and a vector y ∈ CM, respectively, complete the following JULIA function so that it efficiently computes the Euclidean distance between y and the point closest to y in R(B). For full credit, do not use any SVD operations.

function dis_range(B, y)

16. Let x, y and z denote three nonzero orthogonal vectors of the same dimension. Complete the following JULIA function so that it returns the Moore-Penrose pseudoinverse of the matrix A = xx′ + yy′ + zz′ , computed as efficiently as possible.

(Do not call svd or pinv .)

function pinvorth(x, y, z)