Derivatives of Multilinear Functions of Matrices

TL;DR

This paper surveys recent advances in understanding higher order derivatives of matrix functions like determinant, permanent, and tensor powers, and their applications in deriving perturbation bounds.

Contribution

It provides a comprehensive overview of recent results on derivatives of matrix functions and their norms, with new bounds derived via Taylor's theorem.

Findings

Higher order derivatives of matrix functions are crucial for perturbation analysis.

Recent bounds on derivatives improve understanding of matrix function sensitivities.

Taylor's theorem enables precise higher order perturbation bounds for these functions.

Abstract

Perturbation or error bounds of functions have been of great interest for a long time. If the functions are differentiable, then the mean value theorem and Taylor's theorem come handy for this purpose. While the former is useful in estimating $\|f(A+X)-f(A)\|$ in terms of $\|X\|$ and requires the norms of the first derivative of the function, the latter is useful in computing higher order perturbation bounds and needs norms of the higher order derivatives of the function. In the study of matrices, determinant is an important function. Other scalar valued functions like eigenvalues and coefficients of characteristic polynomial are also well studied. Another interesting function of this category is the permanent, which is an analogue of the determinant in matrix theory. More generally, there are operator valued functions like tensor powers, antisymmetric tensor powers and symmetric tensor powers which have gained importance in the past. In this article, we give a survey of the recent work on the higher order derivatives of these functions and their norms. Using Taylor's theorem, higher order perturbation bounds are obtained. Some of these results are very recent and their detailed proofs will appear elsewhere.