ECM5901 Assignment #1 due March 1 n×n † † 1. A matrix is Hermitian A ∈ C ∗ if A = A, where A is the conjugate transpose of A. For example, ∗ a b a c if A = , then A† = ∗ , where a∗ is the conjugate of a ∈ C. A Hermitian matrix allows c d b d∗ an eigenvalue (spectral) decomposition A = U ΛU † , where U ∈ Cn×n is a unitary matrix such that U † U = U U † = I and Λ = diag (λ1 , . . . , λn ) is a diagonal matrix with entries that are eigenvalues of A. (In fact, a normal matrix B such that BB † = B † B allows a spectral decomposition.) (a) Show that a matrix A ∈ Cn×n is Hermitian if and only if A is normal and it has real eigenvalues. (10%) (b) Suppose that A is Hermitian. Show that x† Ax ≥ 0 for any x ∈ Cn if and only if λi ≥ 0 for i = 1, . . . , n. (A is called positive semidefinite if all the eigenvalues λi are nonnegative.) (10%) Pn 2. The trace of a square matrix A = [aij ] ∈ Cn×n is defined as trA = i=1 aii . Show that: (a) tr(AB) = tr(BA) for A, B ∈ Cn×n . (5%) (b) tr(cA + B) = ctr(A) + tr(B) for A, B ∈ Cn×n and c ∈ R. (5%) 3. For a symmetric matrix A ∈ Rn×n , it has an eigenvalue (spectral) decomposition A = QΛQT , where Q ∈ Rn×n is an orthogonal matrix such that QT Q = QQT = I and Λ = diag (λ1 , . . . , λn ) is a diagonal matrix with entries that are eigenvalues of A. (a) Show that trA = n X λi . i=1 (5%) (b) Let λmax = max{λ1 , . . . , λn } and λmin = min{λ1 , . . . , λn }. Show that λmin xT x ≤ xT Ax ≤ λmax xT x for any nonzero vector x ∈ Rn . (10%) 4. Let ⟨ , ⟩ be the inner product on Cn . Prove the Cauchy-Schwarz inequality that for x, y ∈ Cn , 2 |⟨x, y⟩| ≤ ⟨x, x⟩⟨y, y⟩. (10%) 5. Verify that the ℓ1 -norm on Cn defined by ∥x∥1 = Pn i=1 |xi | for x ∈ Cn is a norm on Cn . (10%) 6. Show that the dual norm of ℓ1 -norm is the ℓ∞ -norm. (10%) 7. Verify that that Frobenious inner product on Rm×n defined by ⟨ X, Y ⟩F = tr(X T Y ) for X, Y ∈ Rm×n is an inner product. (10%) 8. Read Appendix A.4. Let A ∈ Rm×n and b ∈ Rm . Consider f : Rn → R defined by f (x) = ||Ax − b||22 for x ∈ Rn . Show that (a) ∇f (x) = 2AT (Ax − b). (10%) (b) ∇2 f (x) = 2AT A. (10%) 1