6. Let A and B be n x n hermitian matrices. Suppose thatA² = InB² = In(1)and[A, B] + = AB + BA=0n.(2)Let x € Cn be normalized, i.e. ||x|| = 1. Here x is considered as a columnvector.(i) Show that(x* Ax)² + (x* Bx)² ≤ 1.(3)(ii) Give an example for the matrices A and B.

Answered: Image /qna-images/answer/f398e37b-4ce0-4a2a-87a1-4b792b26fca9.jpg

Answered: 6. Let A and B be n x n hermitian…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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3. If A is a nonsingular matrix, then A' is nonsingular and its inverse is (A')1 = (A) '. Prove it!
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7. Let A be an n x n Hermitian matrix. (a) Show that (Au, v) = (u, Av) for any n-dim vectors u and v. Here (, ·) denotes the inner product. (Hint: rewrite the inner product as matrix multiplication.) (b) Let x be an eigenvector associated to the eigenvalue A. Based on part (a), show that A(x, x) = X(x, x). (c) As in part (b), show that = X; that is, the eigenvalue A is real. 1
п> 2, neN For every , let A,B,C,D be square А, В, С, DeM, (R) matrices such that AC – BD = InandAD + BC = 0n Prove that det(AC') > 0
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4. Let A be a nonzero 3 x 3 matrix such that A? = 0. (i) By showing the column space of A is contained in the null space of A, show that rank(A) < nullity (A). (ii) Using your answer from part (i), show that rank(A) = 1. %3D

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6. Let A and B be n x n hermitian matrices. Suppose that A² = Ins B² = In (1) and [A, B]+ = AB + BA=0n. (2) Let x € Cn be normalized, i.e. ||x|| = 1. Here x is considered as a column vector. (i) Show that (x* Ax)² + (x* Bx)² ≤ 1. (3) (ii) Give an example for the matrices A and B.