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Problem 7 (8 points). Frobenius norm and its monotonicity.

Given A E Rmxn, let

||A||F

=

m n

ΣΣα;,

i=1 j=1

known as the Frobenius norm. Further, let (A, B) F = trace (ATB), with A, B e Rmxn known

as the Frobenius inner product.

(a) (2 points). Prove that || ||F is a norm in Rmxn. Hint: You may use the fact that the usual

Euclidean norm ||(₁,. ,n)||2 = √x+. + x² is indeed a norm.

(b) (2 points). Prove that (,) is an inner product in Rmxn.

(c) (1 points). Prove that || A+ B|| = ||A||2 + 2(A, B) p + || B|| 21.

(d) (3 points). Suppose that m = n, A, B 0, and AB (i. E. , B-A0). Prove that

||A||F ≤ ||B||F (this is known as monotonicity of || ||F). Hint: If X 0, then there exists

another matrix Y0 such that Y² = X. Such Y is not unique in general, but we may denote it as X¹/2

Note: Please solve (a), (b), (c), (d) in the order that they appear

Sagot :

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