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 Fréchet 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.
From: Vanni Noferini [view email]
[v1]
Tue, 9 Dec 2025 09:31:22 UTC (23 KB)
[v2]
Thu, 18 Dec 2025 08:47:17 UTC (26 KB)
[v3]
Sun, 21 Dec 2025 01:13:33 UTC (26 KB)
[v4]
Wed, 6 May 2026 18:07:49 UTC (27 KB)
[v5]
Thu, 18 Jun 2026 09:52:11 UTC (27 KB)
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