The Jordan canonical form of the Fr\'{e}chet derivative of a matrix function and the bivariate Jordan problem

TL;DR

This paper characterizes the Jordan canonical form of the Fréchet derivative of matrix functions and extends the analysis to linear combinations of Kronecker products, providing new theoretical insights.

Contribution

It determines the Jordan form of the Fréchet derivative of matrix functions and offers partial results for a generalized problem involving Kronecker products.

Findings

Explicit Jordan form of the Fréchet derivative for matrix functions.

Solution to a problem posed by Higham on matrix functions.

Partial results and bounds for a generalized Kronecker product problem.

Abstract

Let $\mathbb{F}$ be an algebraically closed field of characteristic $0$. Given a square matrix $A \in \mathbb{F}^{n \times n}$ and a polynomial $f \in \mathbb{F}[w]$, we determine the Jordan canonical form of the formal Fr\'{e}chet derivative of $f(A)$, in terms of that of $A$ and of $f$. When $\mathbb{F}\subseteq \mathbb{C}$, via Hermite interpolation, our result provides a solution to [N.J. Higham, \emph{Functions of Matrices: Theory and Computation}, Research Problem 3.11]. A generalization consists of finding the Jordan canonical form of linear combinations of Kronecker products of powers of two square matrices, i.e., $\sum_{i,j} a_{ij} (X^i \otimes Y^j)$. For this generalization, we provide some new partial results, including a partial solution under certain assumptions and general bounds on the number and the sizes of Jordan blocks.