Problem 6 Let A, B, C Rnxn where AB = BA = In and AC = CA = Γη. Show that B-C
3. Let U E Rnxn be an orthogonal matrix, i.c., UTU = UUT-1. Show that for any vector x E Rn LXTL we have |lU 2 2. Thus the 2-norm of a vector does not change when it is multiplied by an orthogonal matrix. 3. Let U E Rnxn be an orthogonal matrix, i.c., UTU = UUT-1. Show that for any vector x E Rn LXTL we have |lU 2 2. Thus the 2-norm of a vector does not change...
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...
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...
Problem #2 (a) Prove that 2.At At At where A E Rnxn (b) If (λί,ui), i-1, 2, . .. , n, are the eigenvalue-eigenvector pairs of A Rnxn, what are the eigenvalues and eigenvectors of e? Prove your answer Problem #2 (a) Prove that 2.At At At where A E Rnxn (b) If (λί,ui), i-1, 2, . .. , n, are the eigenvalue-eigenvector pairs of A Rnxn, what are the eigenvalues and eigenvectors of e? Prove your answer
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...
3. [2+2pt] Let n > 2. Consider a matrix A E Rnxn for which every leading principal submatrix of order less than n is non-singular. (a) Show that A can be factored in the form A = LDU, where Le Rnxn is unit lower triangular, D e Rnxn is diagonal and U E Rnxn is unit upper triangular. (b) If the factorization A = LU is known, where L is unit lower triangular and U is upper triangular, show how...
6. [20] Let A, B e Cnxn such that A2 = A and B2 = B. Prove that if (A + B)2 = A + B, then AB is the zero matrix 0 Rnxn.
4.[10 points] Let A be the language over the alphabet E-(a, b} defined by regular expression (ab U b)*a U b. Give an NFA that recognizes A. Draw an NFA for A here. 4.[10 points] Let A be the language over the alphabet E-(a, b} defined by regular expression (ab U b)*a U b. Give an NFA that recognizes A. Draw an NFA for A here.
Problem 5.4 (10 points) Let (Sn)n-01. be a simple, symmetric random walk with starting value So-s e R. (a) Show that ES for alln0 b) Show that ElSn+1 Sn] Sn for 0. (c)Suppose that (Sn)n-0,12,. . denotes the profit and loss from $1 bets of a gambler with initial capital So-s who is repeatedly playing a fair game with 50% chances to win or lose her stake. What are the interpretations of the results in (a) and (b)? Problem 5.4...