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arxiv: 2512.08399 · v4 · submitted 2025-12-09 · 🧮 math.RA

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

Vanni Noferini This is my paper

Pith reviewed 2026-05-16 23:42 UTC · model grok-4.3

classification 🧮 math.RA MSC 15A21
keywords Jordan canonical formFréchet derivativematrix functionsKronecker productsHermite interpolationbivariate Jordan problemalgebraically closed fields
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The pith

The Jordan canonical form of the Fréchet derivative of f(A) is determined by the Jordan form of A and the polynomial f.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines the Jordan canonical form of the formal Fréchet derivative of f(A) for a polynomial f and square matrix A, expressing it directly from the Jordan structure of A and the action of f. This matters because understanding the derivative operator helps analyze how matrix functions respond to perturbations, with applications in numerical linear algebra and control theory. When the field is the complex numbers, the result solves an open research problem on this topic using Hermite interpolation. A related bivariate problem on Jordan forms of sums of Kronecker products receives partial solutions and bounds on block sizes.

Core claim

Given an algebraically closed field F of characteristic zero, a square matrix A in F^{n x n} and a polynomial f in F[w], the Jordan canonical form of the formal Fréchet derivative of f(A) is determined in terms of the Jordan canonical form of A and of f. The result extends via Hermite interpolation to solve the corresponding problem over the complexes, and partial results are given for the Jordan form of linear combinations of the form sum a_{ij} (X^i ⊗ Y^j).

What carries the argument

The formal Fréchet derivative of the map sending a matrix X to f(X), whose Jordan canonical form is constructed from the Jordan blocks of A using f and its derivatives at the eigenvalues of A.

If this is right

  • The Jordan form of the derivative operator can be constructed explicitly from the Jordan blocks of A and the derivatives of f.
  • It solves Research Problem 3.11 from Higham's book on functions of matrices for the complex case.
  • Bounds on the number and sizes of Jordan blocks are provided for the bivariate Kronecker product case.
  • Partial explicit solutions are available for the bivariate problem under specific assumptions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical algorithms for matrix function derivatives can use this structure for efficiency.
  • The approach may generalize to analytic matrix functions beyond polynomials.
  • The bivariate results could help analyze stability in systems with multiple matrix variables.

Load-bearing premise

The field is algebraically closed of characteristic zero and f is a polynomial.

What would settle it

For a 2x2 Jordan block matrix A with a single eigenvalue and a quadratic polynomial f, explicitly compute the Fréchet derivative operator on 4x4 matrices and check whether its Jordan blocks match the sizes and numbers predicted from A's structure and f's derivatives.

Figures

Figures reproduced from arXiv: 2512.08399 by Vanni Noferini.

Figure 1
Figure 1. Figure 1: Illustration of a claim within the proof of Proposition 3.11 for [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript determines the Jordan canonical form of the formal Fréchet derivative of a polynomial matrix function f(A) for an n×n matrix A over an algebraically closed field of characteristic zero, expressing the result directly in terms of the Jordan structure of A and the divided differences induced by f. It reduces the problem to single Jordan blocks, applies the known action of the derivative on such blocks, and assembles the global form; when the field is a subfield of the complexes, Hermite interpolation yields a solution to Higham’s Research Problem 3.11. Partial results and bounds on block numbers and sizes are also given for the bivariate generalization consisting of linear combinations of Kronecker products of powers of two matrices.

Significance. If the explicit construction holds, the work supplies a complete, constructive answer to a long-standing open problem in matrix-function theory and supplies concrete data (Jordan block sizes and eigenvalues) that can be used in both theoretical proofs and numerical algorithms for Fréchet derivatives. The bivariate bounds, although limited to special coefficient and spectral regimes, are new and may serve as a starting point for further progress on the general Kronecker-sum Jordan problem.

minor comments (2)
  1. [Main theorem section] The reduction to a single Jordan block (presumably in the main theorem statement) would be easier to follow if the precise block-wise formula for the Fréchet derivative were displayed as a separate displayed equation before the global assembly argument.
  2. [Bivariate results] In the bivariate section the precise hypotheses on the coefficients a_{ij} and on the spectra of X and Y that permit the partial Jordan-form description should be stated as a numbered assumption or hypothesis immediately before the corresponding theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The report correctly summarizes our main contribution: an explicit determination of the Jordan canonical form of the Fréchet derivative of a polynomial matrix function f(A) in terms of the Jordan structure of A and the divided differences of f. We are pleased that the work is viewed as supplying a constructive answer to Higham's Research Problem 3.11 and providing new partial results for the bivariate Kronecker case. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation reduces the problem to the action of the Fréchet derivative on individual Jordan blocks of A, applies the standard divided-difference formula for that action (a known result from matrix function theory), and assembles the global Jordan structure via classical similarity transformations and Hermite interpolation. No equation or step is defined in terms of its own output, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on a self-citation chain. The bivariate partial results are explicitly qualified by extra assumptions and do not claim a complete closed-form result. The central claim therefore remains an independent explicit construction built on external classical facts.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivation rests on classical linear algebra over the stated field and standard interpolation techniques without introducing new free parameters or postulated entities.

axioms (1)
  • domain assumption F is an algebraically closed field of characteristic 0
    This is the explicit setting stated for the main Jordan form result.

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Reference graph

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