Page 297 - Matrix Analysis & Applied Linear Algebra

5.3 Inner-Product Spaces 293

5.3.5. For n × n matrices A and B, explain why each of the following in-
equalities is valid.
2
∗
(a) |trace (B)| ≤ n [trace (B B)] .
2 T
(b) trace B ≤ trace B B for real matrices.
T T
trace A A + trace B B

T
(c) trace A B ≤ for real matrices.
2
5.3.6. Extend the proof given on p. 290 concerning the parallelogram identity
(5.3.7) to include complex spaces. Hint: If V is a complex space with
a norm that satisfies the parallelogram identity, let

2 2
x + y − x − y
x y = ,
r 4
and prove that

x y = x y +i ix y (the polarization identity) (5.3.10)
r r
is an inner product on V.

n
5.3.7. Explain why there does not exist an inner product on C (n ≥ 2) such

that = .
∞
n×n
5.3.8. Explain why the Frobenius matrix norm on C must satisfy the par-
allelogram identity.

5.3.9. For n ≥ 2, is either the matrix 1-, 2-, or ∞-norm generated by an inner
n×n
product on C ?