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.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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6. Let A and B be n x n hermitian matrices. Suppose that
A² = In
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.
Transcribed Image Text:6. Let A and B be n x n hermitian matrices. Suppose that A² = In 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.
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