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The Jordan canonical form of the Fréchet derivative of a matrix function and the bivariate Jordan problem.

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Τίτλος: The Jordan canonical form of the Fréchet derivative of a matrix function and the bivariate Jordan problem.
Συγγραφείς: Noferini, Vanni1 (AUTHOR) vanni.noferini@aalto.fi
Πηγή: Linear Algebra & its Applications. Oct2026, Vol. 747, p48-76. 29p.
Θεματικοί όροι: *Matrix functions, *Kronecker products, *Derivatives (Mathematics), *Interpolation, *Matrix decomposition
Περίληψη: Let F be an algebraically closed field of characteristic 0. Given a square matrix A ∈ F n × n and a polynomial f ∈ 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 F ⊆ C , via Hermite interpolation, our result provides a solution to N.J. Higham (2008) [7, 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., ∑ i , j a i j (X i ⊗ 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. [ABSTRACT FROM AUTHOR]
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  Data: Let F be an algebraically closed field of characteristic 0. Given a square matrix A ∈ F n × n and a polynomial f ∈ 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 F ⊆ C , via Hermite interpolation, our result provides a solution to N.J. Higham (2008) [7, 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., ∑ i , j a i j (X i ⊗ 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. [ABSTRACT FROM AUTHOR]
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      – SubjectFull: Derivatives (Mathematics)
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              Text: Oct2026
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