On directional derivatives of trace functionals of the form A↦Tr(Pf(A))

• Published in: Linear algebra and its applications Vol. 569; pp. 62 - 77
• Main Author: Girard, Mark W.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 15.05.2019; American Elsevier Company, Inc
• Subjects: Computation; Derivatives; Eigenvalue perturbation; Linear algebra; Mathematical analysis; Mathematical programming; Matrix methods; Positive semidefinite Hermitian matrices; Quantum relative entropy; Trace functional; Eigenvalue perturbation; 15A99; Positive semidefinite Hermitian matrices; Quantum relative entropy; 15B48; Trace functional; 15A60
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2019.01.012

Given a function f:(0,∞)→R and a positive semidefinite n×n matrix P, one may define a trace functional on positive definite n×n matrices as A↦Tr(Pf(A)). For differentiable functions f, the function A↦Tr(Pf(A)) is differentiable at all positive definite matrices A. Under certain continuity conditions on f, this function may be extended to certain non-positive-definite matrices A, and the directional derivatives of Tr(Pf(A)) may be computed there. This note presents conditions for these directional derivatives to exist and computes them. These conditions hold for the function f(x)=log⁡(x) and for the functions fp(x)=xp for all p>−1. The derivatives of the corresponding trace functionals are computed here, and an alternative derivation of the directional derivatives using integral representations is provided.