2. Let A e cnxn and A BiC, where B,C E Rnxn and i -I. Denote B -C (a) Show that A is unitary if a...
Question B 7. (a) Let -1 0 0 (i) Find a unitary matrix U such that M-UDU where D is a diagonal matrix. 10 marks] (i) Compute the Frobenius norm of M, i.e., where (A, B) = trace(B·A). [4 marks] 3 marks] (iii) What is NM-illp? (b) Let H be an n × n complex matrix (6) What does it mean to say that H is positive semidefinite. (il) Show that H is positive semidefinite and Hermitian if and only...
A. (For Math 603 students only) Consider a symmetric and positive definite matrix A Rnxn and let λ'nin(A) and Xmax(A) be the minimal and maximal real eigenvalues of A respectively. Show that Suggested readings: Sections 7.2, 7.5, 7.6 A. (For Math 603 students only) Consider a symmetric and positive definite matrix A Rnxn and let λ'nin(A) and Xmax(A) be the minimal and maximal real eigenvalues of A respectively. Show that Suggested readings: Sections 7.2, 7.5, 7.6
A real symmetric matrix B e Rnxn (i.e. BT = B) is said to be positive definite if all of its eigenvalues 11, 12, ..., In are positive. (Recall that is an eigenvalue of B if and only if there exits a nonzero vector t such that Bt = it). Show that B-1 is also positive definite. That is, you need to show that all the eigenvalues of B-1 are also positive. (Hint: consider equation Bt; = liti for all...
LINEAR ALGEBRA Problem 10.4 (Math 6435). Let A = [a] e Cnxn and assume that A is Hermitian (1) Prove that the diagonal entries of A (i.e., ai for 1 < i < n) are real numbers. (2) Prove that, for every BE Cxm, BHAB is a Hermitian matrix of size m x m Hint. (1) A complex number is real if and only if it coincides with its conjugate (2) Observe the equations (XY)# = Y#x¥ and (X#)H =...
Question 4: Eigenvalue Theory 2 Let A Cnxn. For each of the following statements show that it is true or give a counterexample to show that it is false (a) If λ is an eigenvalue of A, and μ є Cn then λ-μ is an eigenvalue of A-1 (b) If A is real and λ is an eigenvalue of A then so is-λ. (c) If A is real and λ is an eigenvalue of A, then so is λ. (d)...
3, (a) [5 marks] what does it mean for A E Rnxn to be (i) symmetric? (ii) orthogonal? (ii) diagonalisable? (iv) orthogonally diagonalisable? (b) [4 marks] Suppose that A ERn is orthogonally diagonalisable. Prove that A is symmetric. (c) [11 marks] Let A be the matrix 6 -2 Show that the eigenvalues are 7 and -6. Show that any corresponding eigenvectors vi and v2 are orthogonal with respect to the Euclidean inner product (d) [5 marks] Hence prove that the...
(f) Let A be symmetric square matrix of order n. Show that there exists an orthogonal matrix P such that PT AP is a diagonal matrix Hint : UseLO and Problem EK〗 (g) Let A be a square matrix and Rn × Rn → Rn is defined by: UCTION E AND MES FOR THE la(x, y) = хтАУ (i) Show that I is symmetric, ie, 14(x,y) = 1a(y, x), if a d Only if. A is symmetric (ii) Show that...
I need help with a, b, and c. 7.Let A be ann x n real symmetric invertible matrix, let B Rt and C E R. Define f:R R by 2 a. Give f (a) c. Give f"(x) d. Prove that if A is positive definite and u is the critical point of f, then f(u) < f(x) for all x E Rn where x Prove that if A is negative definite and u is the critical point of f, then...
12. (True/False) (a) Let AE Rm*n . Then R(A) (b) Let AERm*n. Then N(A) is isomorphic to N(AT) (c) We define < A. B > = Tr (BTA ) where A, B E Rnxn . is isomorphic to R(A Then 〈 . , . 〉 is an inner product on Rmxn. (d) Consider a periodic-function space V with period of 1 sec. Define an inner product on V by <f,a>= f(t )a (t ) dt. Then cos 2 π t...
a through e is considered one question. 7.Let A be ann x n real symmetric invertible matrix, let B Rt and C E R. Define f:R R by 2 a. Give f (a) c. Give f"(x) d. Prove that if A is positive definite and u is the critical point of f, then f(u) < f(x) for all x E Rn where x Prove that if A is negative definite and u is the critical point of f, then f(u)...