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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 and only if M is orthogonal. (b) Show that A is Hermitian positive definite if and only if M is symmetric positive definite. (c) Suppose A is Hermitian positive definite. Design an algorithm for solving Ar-busing real arithmetic only 2. Let A e cnxn and A BiC, where B,C E Rnxn and i -I. Denote...
(c) If A is a square matrix and A2 = 0,then A = 0. (d) Let A, B be two square matrices. If (A + B) 2 = A2 + 2AB + B2 , then AB = BA.
e) Use the triangle inequality to prove that (ac + bd)2 (a2 + b2)(c2 + d2) for all a, b, c, d e R. Total: [20 marks] e) Use the triangle inequality to prove that (ac + bd)2 (a2 + b2)(c2 + d2) for all a, b, c, d e R. Total: [20 marks]
(a) Let P(B1∩B2)>0, and A1∪A2⊂B1∩B2. Then show that P(A1|B1).P(A2|B2)=P(A1|B2).P(A2|B1). (b) Let A and B1 be independent; similarly, let A and B2 be independent. Show that in this case, A and B1∪B2 are independent if and only if A and B1∩B2 are independent. (c) Given P(A) = 0.42,P(B) = 0.25, and P(A∩B) = 0.17, find (i)P(A∪B) ; (ii)P(A∩Bc) ; (iii)P(Ac∩Bc) ; (iv)P(Ac|Bc).
2. (a) Let P(Bin B2) > 0, and AUA, CBin B2. Then show that P(A/B).P (A2|B2) = P(A|B2).P (A2|Bi). (b) Let A and Bbe independent; similarly, let A and B, be independent. Show that in this case, A and B U B2 are independent if and only if A and Bin B2 are independent (c) Given P(A) = 0.42, P(B) = 0.25, and P(An B) = 0.17, find (i) P (AUB); (ii) P(An B°); (iii) P(A n B); (iv) P(...
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 =...
Prove that ab ≤ 1/2(a2 + b2) for any real numbers a and b.
2. Consider the reaction A2+ B2 = 2AB. If the initial concentration of both A2 and B2 is 4.0 M. and after 10 minutes the reaction appears to stop. The concentration of [A2) is now 2.0M. a. Draw a graph of [A2) vs. time, over the span of 20 minutes. 09. (M concertuohon 5 15 20 time (min) b. On the same axes draw a graph of [B2] v time, over a span of 20 minutes. c. On the same...
2. Consider the reaction A2 + B2 ⇌ 2AB. If the initial concentration of both A2 and B2 is 4.0 M, and after 10 minutes the reaction appears to stop. The concentration of [A2] is now 2.0M. a. Draw a graph of [A2] vs. time, over the span of 20 minutes. b. On the same axes draw a graph of [B2] v time, over a span of 20 minutes. c. On the same axes draw a graph of [AB] v...
Exercise 6 (6.4.35, p.452) Let A e Cnxn, and let S be a k-dimensional subspace of C". Then a vector ve S is called a Ritz vector of A from S if and only if there is a pie C such that the Rayleigh-Ritz-Galerkin condition Av – uv Is holds, that is, (Av – uv, s) = 0 for all s E S. The scalar u is called the Ritz value of A associated with v. Let 91, ...,qk be...