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Let U E Mat(n; C) be a unitary matrix. Show that (UzUy) = (y) for any...
8) Let U be a unitary matrix. Show that: a. U is normal. b. Ifa is an eigenvalue of U then N= 1 (Hint: 1 Uxl =1 xl for every x e C”.
8) Let U be a unitary matrix. Show that: a. U is normal. b. If2 is an eigenvalue of Uthen = 1 (Hint: Uxl =1 xl for every x € C".
Let V be R, with thestandard inner product. If is a unitary operator on V, show that the matrix of U in the standard ordered basis is either cos θ -sin θ sin θ cos θ cos θ sin θ for some real θ, 0-θ < 2T. Let Us be the linear operator corresponding to the first matrix, i.e., Ue is rotation through the angle . Now convince yourself that every unitary operator on V is either a rotation, or...
8) Let U be a unitary matrix. Show that: a. U is normal. b. If 2 is an eigenvalue of Uthen 2=1 (Hint: 1 Uxl l xl for every xC".
Let A be an m × n matrix, let x Rn and let 0 be the zero vector in Rm. (a) Let u, v є Rn be any two solutions of Ax 0, and let c E R. Use the properties of matrix-vector multiplication to show that u+v and cu are also solutions of Ax O. (b) Extend the result of (a) to show that the linear combination cu + dv is a solution of Ax 0 for any c,d...
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...
6.3 (Adjugate matrix) a) Let A E GLn(K). Use Cramer's rule to show that A-1 = data adj A without using Lemma 4.5.17. b) Let A Knxn be an upper triangular matrix (i.e. ajj = 0 (1 <j<i<n). Show that adj A is an upper triangular matrix. c) Let Znxn := {A € RNXN | aij € Z (i, j = 1, ..., n)}. Show that U := {A € Znxn | det(A) = 1} is a group with respect...
Please help with this multivariable calculus problem in manifold! Show that the set U(n) of unitary n x n - matrices (those n x n - matrices A with entries in complex number such that BTA=the identity matrix, where B is the matrix with entries the complex conjugates of those of A, i.e. if the 1st row and 1st column entry of A is i, then the 1st row and 1st column entry of B is -i) is a manifod....
4. Let T be the time reversal operator. Show that T=U*K where U is the unitary operator and K is the Operator of conjugation . Use the relation TST --S describing time reversed spin operator S to show that T-UK where U = 1 sigmay
Problem 4. Let GL2(R) be the vector space of 2 x 2 square matrices with usual matrix addition and scalar multiplication, and Wー State the incorrect statement from the following five 1. W is a subspace of GL2(R) with basis 2. W -Ker f, where GL2(R) R is the linear transformation defined by: 3. Given the basis B in option1. coordB( 23(1,2,2) 4. GC2(R)-W + V, where: 5. Given the basis B in option1. coordB( 2 3 (1,2,3) Problem 5....