ABSTRACT

Let M n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351142120/c81116b1-56b8-4322-b026-4912085dda0d/content/eq546.tif"/> be the set of all n–by–n matrices on ℂ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351142120/c81116b1-56b8-4322-b026-4912085dda0d/content/eq547.tif"/> with an identity matrix I. A matrix A ∈ M n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351142120/c81116b1-56b8-4322-b026-4912085dda0d/content/eq548.tif"/> is said to be positive semi-definite if 〈Ax, x〉 ≥ 0 holds for all x ∈ ℂ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351142120/c81116b1-56b8-4322-b026-4912085dda0d/content/eq549.tif"/> . Moreover A ∈ M n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351142120/c81116b1-56b8-4322-b026-4912085dda0d/content/eq550.tif"/> is called positive definite if A is positive semi-definite and invertible. We use a notation A ≥ 0 (resp. A > 0) for positive semi-definite (resp. positive definite) matrices. In this chapter, P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351142120/c81116b1-56b8-4322-b026-4912085dda0d/content/eq551.tif"/> Sn and P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351142120/c81116b1-56b8-4322-b026-4912085dda0d/content/eq552.tif"/> n denote the sets of all positive semi-definite and positive definite matrices, respectively. For Hermitian A, B ∈ M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351142120/c81116b1-56b8-4322-b026-4912085dda0d/content/eq553.tif"/> n , A ≤ B is defined by 0 ≤ B – A. A real-valued function f defined on a real interval I is said to be matrix monotone if f(A) ≤ f(B) holds for all Hermitian A, B ∈ M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351142120/c81116b1-56b8-4322-b026-4912085dda0d/content/eq554.tif"/> n such that A ≤ B and σ(A), σ(B) ⊂ I, where σ(T) means the spectrum of a T ∈ M https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351142120/c81116b1-56b8-4322-b026-4912085dda0d/content/eq555.tif"/> n . Here f(A) is defined by f ( A ) = ∑ i = 1 n f ( λ i ) P i , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351142120/c81116b1-56b8-4322-b026-4912085dda0d/content/eq556.tif"/>