arXiv subjects

Vanni Noferini

Publications and source records attributed to Vanni Noferini.

At least 19 recordsLinked to original sources

arXiv2026

Matrix Nearness Problems Revisited

Given a nonempty set of matrices $\mathcal{E} \subseteq \mathbb{C}^{m \times n}$ and a matrix $A \not\in\mathcal{E}$, a matrix nearness problem (or matrix distance problem) consists in finding a matrix in $\mathcal{E}$ that is closest to $A$, in some norm, as well as the corresponding distance. This article is a survey of matrix nearness problems and of numerical algorithms for their solution. We pay special attention to modern related topics, such as Riemannian optimization and pseudospectra, that were not discussed in depth in previous reviews about this topic. Besides collecting known facts, we also prove some foundational results that, at least in the present form, we could not locate in the literature.

math.NA↗

arXiv2026

Matrix nearness problems: Do real inputs admit real solutions?

Given a real square matrix $A$ and a nonempty closed target set of complex matrices $\mathcal{E}$, invariant by complex conjugation and containing real matrices, does the subset of $\mathcal{E}$ consisting of the matrices nearest to $A$ in the Frobenius distance always contain a real matrix? We give negative answers when $\mathcal{E}$ is either the set of normal matrices (solving a question posed by N. Higham) or the set of matrices whose eigenvalues lie in the closed left half-plane (solving a question posed by the first author and F. Poloni). For the latter problem, we also argue that the ratio between the real and complex distances is unbounded in every dimension $n\geq 3$, and this holds for every distance induced by a unitarily invariant norm. For both problems and $n \geq 3$, we show that a real input $A$ uniformly drawn from the unit sphere $\|A\|_F=1$ has no real minimizer over $\mathcal{E}$ with probability strictly between $0$ and $1$. Beyond these two cases, we prove for an arbitrary nonempty closed target sets $\mathcal{E}$ that, for almost every real input, either $\mathcal{E}$ contains a unique nearest matrix which is real, or every nearest matrix in $\mathcal{E}$ is nonreal.

math.NA↗

arXiv2026

Invariant factors from products of cyclotomic polynomials, with applications

Let $f,g\in\mathbb{Z}[t]$ be products of cyclotomic polynomials such that either $g/\gcd(f,g)$ or $f/\gcd(f,g)$ (or both) is squarefree. We give explicit formulae for the invariant factors of $f(C_g)$, where $C_g$ is the companion matrix of $g$, expressing them in terms of the indices and multiplicities of the cyclotomic factors of $f$ and $g$. A more compact formula holds for the circulant case $g(t)=t^n-1$, assuming that every cyclotomic factor shared by $f$ and $g$ has multiplicity $1$ in $f$. We discuss applications to several topics, including abelianizations of cyclically presented groups, homology groups of cyclic branched covers of knots, adjacency matrices and shifted graph Laplacians, $K$-theory, and Smith forms of matrix powers. For example, for two families of hyperbolic knots we give the first integral homology in every covering degree; or, when $g$ is squarefree, we determine the least eventual period of the Smith forms of powers of $f(C_g)$.

math.RA↗

arXiv2026

Rosenbrock's Theorem characterizes Prüfer domains

Under coprimality assumptions on certain submatrices, Rosenbrock's Theorem relates the Smith form of a matrix $P$ over an elementary divisor domain $\mathcal{R}$ to the Smith-McMillan form of a matrix $G$ over the field of fractions of $\mathcal{R}$, where $G$ is a Schur complement in $P$. If $\mathcal{R}$ is not an elementary divisor domain, Rosenbrock's Theorem is not directly applicable in its original form, because not every matrix is unimodularly equivalent to a matrix in Smith form. In this paper, we state an ideal-theoretic version of Rosenbrock's Theorem that is meaningful over any integral domain, and we show that it is equivalent to the classic formulation over an elementary divisor domain. Moreover, we give a characterization of Prüfer domains as those integral domains over which the ideal-theoretic version of Rosenbrock's Theorem holds for every matrix satisfying the assumptions. In particular, the theorem does not hold for every admissible matrix over $\mathbb{C}[x_1,\dots,x_d]$ when $d \geq 2$. However, when $d\leq 3$, it holds for square matrices in a nonempty Zariski-open subset of the coefficient space, having fixed the block sizes and a total degree bound. Finally, we prove that, if $\mathcal{R}$ is an integral domain such that every right invertible matrix can be completed to a unimodular matrix, then every matrix $P$ that satisfies the assumptions of the ideal-theoretic Rosenbrock's Theorem and realizes the same Schur complement $G$ shares the same ideal-theoretic generalization of the Smith form.

math.AC↗

arXiv2026

Closest Normal Matrix Found Again Using Riemannian Optimization

We propose an approach based on Riemannian optimization to compute a nearest normal matrix to a given one. The problem can be formulated as the minimization of a smooth function either on the manifold $U(n)$ of unitary matrices of size n or on the flag manifold $U (n)/U (1)^n$. The flag manifold is particularly suitable for theoretical analysis; we characterize the global maximum of the objective function and prove that, for generic inputs, its local minimizers are finitely many and isolated; in turn, this implies the original nearest normal matrix problem generically has finitely many local minimizers, all with distinct eigenvalues. We also develop a Riemannian trust-region method that improves substantially on classical algorithms and can handle considerably larger matrices, as well as a variant for computing the nearest real normal matrix. The paper is complemented by extensive numerical experiments.

math.NA↗

arXiv2026

Nearest matrix with multiple eigenvalues by Riemannian optimization

Given a square complex matrix $A$, we tackle the problem of finding the nearest matrix with multiple eigenvalues or, equivalently when $A$ had distinct eigenvalues, the nearest defective matrix. To this goal, we extend the general framework described in [M. Gnazzo, V. Noferini, L. Nyman, F. Poloni, \emph{Riemann-Oracle: A general-purpose Riemannian optimizer to solve nearness problems in matrix theory}, Found. Comput. Math., To appear] and based on variable projection and Riemannian optimization, allowing the ambient manifold to simultaneously track left and right eigenvectors. Our method also allows us to impose arbitrary complex-linear constraints on either the perturbation or the perturbed matrix; this can be useful to study structured eigenvalue condition numbers. We present numerical experiments, comparing with preexisting algorithms.

math.NA↗

arXiv2026

Fast stability tests for Hermitian matrix polynomials

Assessing the asymptotic stability of linear self-adjoint homogeneous systems of differential-algebraic equations requires testing the Hurwitz stability of the associated Hermitian matrix polynomial $P(λ)$. Tests for known necessary and sufficient conditions rely on linearizations and eigensolvers, solving matrix equations and testing matrix inequalities, or generalized Bézoutians, and scale with either $O(d^2 n^3)$ or $O(d^3n^3)$ complexity, where $d$ and $n$ are the degree and size of $P(λ)$, respectively. We establish several novel sufficient conditions for stability, based on the numerical range of $P(λ)$. Based on the new results, we propose algorithms with $O(d n^3)$ asymptotic complexity. Our methods rely on very efficient core numerical linear algebra routines, such as the Cholesky decomposition of $n \times n$ matrices or the computation of the largest eigenvalue of $n \times n$ definite pencils. Therefore, a significant computational advantage can be expected in favor of the proposed approach even for moderate values of $d$ or $n$, and we verify this with numerical experiments.

math.NA↗

arXiv2026

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

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.

math.RA↗

arXiv2026

Eventual periodicity of the Smith forms of integer matrix powers

We prove that the Smith forms of the powers of an integer square matrix behave in an eventually periodic manner. More precisely, if $\mathrm{SF}(M)$ denotes the Smith form of $M \in \mathbb{Z}^{m \times m}$, then for every $A \in \mathbb{Z}^{m \times m}$ there exist $n_0 \in \mathbb{N}$, an integer $T \geq 1$, and a constant diagonal matrix $D \in \mathbb{Z}^{m \times m}$ such that $n \geq n_0$ implies $\mathrm{SF}(A^{n+T})=D \cdot \mathrm{SF}(A^n)$. This provides an eventually affirmative answer to a conjecture posed in 2013 by R. Bruner. We also show that both $n_0$ and $T$ can be arbitrarily large.

math.NT↗

arXiv2025

The Smith form of Sylvester and Bézout matrices for zero-dimensional ideals

Let $\mathbb{K}$ be a field and let $f,g \in \mathbb{K}[x,y]$ be such that the ideal $\langle f,g \rangle$ is zero-dimensional. We study the Sylvester and Bézout resultant polynomial matrices, built by interpreting $f$ and $g$ as univariate polynomials in $x$ with coefficients in $\mathbb{K}[y]$. We characterize their Smith forms over $\mathbb{K}[y]$ in terms of the dual spaces of differential operators, that were defined and studied by H. M. Möller et al. In particular, if $\mathbb{K}$ is algebraically closed we show that, if the leading coefficients of $f$ and $g$ are coprime over $\mathbb{K}[y]$, then the partial multiplicities of the Sylvester and Bézout resultant matrices coincide with certain integers, that we call Möller indices. These indices are uniquely determined by $\langle f,g \rangle$, and can be easily computed from a Gauss basis, as defined in [M. G. Marinari, H. M. Möller, T. Mora, Trans. Amer. Math. Soc. 348(8):3283--3321, 1996], of the dual spaces. We then generalize this result to the case of common factors in the leading coefficients, which correspond to intersections at $x=\infty$, again describing all the invariant factors of Sylvester and Bézout resultant matrices. As a corollary, this fully characterizes the algebraic multiplicity of all the roots of the resultant $\mathrm{Res}_x(f,g) \in \mathbb{K}[y]$ in terms of the intersection multiplicities for $f$ and $g$, including those arising from infinite intersections. We discuss both algebraic and computational implications of our results.

math.AC↗

arXiv2025

Riemann-Oracle: A general-purpose Riemannian optimizer to solve nearness problems in matrix theory

We propose an extremely versatile approach to address a large family of matrix nearness problems, possibly with additional linear constraints. Our method is based on splitting a matrix nearness problem into two nested optimization problems, of which the inner one can be solved either exactly or cheaply, while the outer one can be recast as an unconstrained optimization task over a smooth real Riemannian manifold. We observe that this paradigm applies to many matrix nearness problems of practical interest appearing in the literature, thus revealing that they are equivalent in this sense to a Riemannian optimization problem. We also show that the objective function to be minimized on the Riemannian manifold can be discontinuous, thus requiring regularization techniques, and we give conditions for this to happen. Finally, we demonstrate the practical applicability of our method by implementing it for a number of matrix nearness problems that are relevant for applications and are currently considered very demanding in practice. Extensive numerical experiments demonstrate that our method often greatly outperforms its predecessors, including algorithms specifically designed for those particular problems.

math.NA↗

arXiv2025

On rank-2 Nonnegative Matrix Factorizations and their variants

We consider the problem of finding the best nonnegative rank-2 approximation of an arbitrary nonnegative matrix. We first revisit the theory, including an explicit parametrization of all possible nonnegative factorizations of a nonnegative matrix of rank 2. Based on this result, we construct a cheaply computable (albeit suboptimal) nonnegative rank-2 approximation for an arbitrary nonnegative matrix input. This can then be used as a starting point for the Alternating Nonnegative Least Squares method to find a nearest approximate nonnegative rank-2 factorization of the input; heuristically, our newly proposed initial value results in both improved computational complexity and enhanced output quality. We provide extensive numerical experiments to support these claims. Motivated by graph-theoretical applications, we also study some variants of the problem, including matrices with symmetry constraints.

math.NA↗

arXiv2025

Finding the nearest $Ω$-stable pencil with Riemannian optimization

This paper considers the problem of finding the nearest $Ω$-stable pencil to a given square pencil $A+xB \in \mathbb{C}^{n \times n}$, where a pencil is called $Ω$-stable if it is regular and all of its eigenvalues belong to the closed set $Ω$. We propose a new method, based on the Schur form of a matrix pair and Riemannian optimization over the manifold $U(n) \times U(n)$, that is, the Cartesian product of the unitary group with itself. While the developed theory holds for any closed set $Ω$, we focus on two cases that are the most common in applications: Hurwitz stability and Schur stability. For these cases, we develop publicly available efficient implementations. Numerical experiments show that the resulting algorithm outperforms existing methods.

math.NA↗

arXiv2024

Efficient computation of \lowercase{$f$}-centralities and nonbacktracking centrality for temporal networks

We discuss efficient computation of $f$-centralities and nonbacktracking centralities for time-evolving networks with nonnegative weights. We present a node-level formula for its combinatorially exact computation which proves to be more tractable than previously existing formulae at edge-level for dense networks. Additionally, we investigate the impact of the addition of a final time frame to such a time-evolving network, analyzing its effect on the resulting nonbacktracking Katz centrality. Finally, we demonstrate by means of computational experiments that the node-level formula presented is much more efficient for dense networks than the previously known edge-level formula. As a tool for our goals, in an appendix of the paper, we develop a spectral theory of matrices whose elements are vectors.

math.NA↗

arXiv2024

Tropical Laurent series, their tropical roots, and localization results for the eigenvalues of nonlinear matrix functions

Tropical roots of tropical polynomials have been previously studied and used to localize roots of classical polynomials and eigenvalues of matrix polynomials. We extend the theory of tropical roots from tropical polynomials to tropical Laurent series. Our proposed definition ensures that, as in the polynomial case, there is a bijection between tropical roots and slopes of the Newton polygon associated with the tropical Laurent series. We show that, unlike in the polynomial case, there may be infinitely many tropical roots; moreover, there can be at most two tropical roots of infinite multiplicity. We then apply the new theory by relating the inner and outer radii of convergence of a classical Laurent series to the behavior of the sequence of tropical roots of its tropicalization. Finally, as a second application, we discuss localization results both for roots of scalar functions that admit a local Laurent series expansion and for nonlinear eigenvalues of regular matrix valued functions that admit a local Laurent series expansion.

math.NA↗

arXiv2024

Smith forms of matrices in Companion Rings, with group theoretic and topological applications

Let $R$ be a commutative ring and $g(t) \in R[t]$ a monic polynomial. The commutative ring of polynomials $f(C_g)$ in the companion matrix $C_g$ of $g(t)$, where $f(t)\in R[t]$, is called the Companion Ring of $g(t)$. Special instances include the rings of circulant matrices, skew-circulant matrices, pseudo-circulant matrices, or lower triangular Toeplitz matrices. When $R$ is an Elementary Divisor Domain, we develop new tools for computing the Smith forms of matrices in Companion Rings. In particular, we obtain a formula for the second last non-zero determinantal divisor, we provide an $f(C_g) \leftrightarrow g(C_f)$ swap theorem, and a composition theorem. When $R$ is a principal ideal domain we also obtain a formula for the number of non-unit invariant factors. By applying these to families of circulant matrices that arise as relation matrices of cyclically presented groups, in many cases we compute the groups' abelianizations. When the group is the fundamental group of a three dimensional manifold, this provides the homology of the manifold. In other cases we obtain lower bounds for the rank of the abelianisation and record consequences for finiteness or solvability of the group, or for the Heegaard genus of a corresponding manifold.

math.GR↗

arXiv2024

Para-Hermitian rational matrices

In this paper we study para-Hermitian rational matrices and the associated structured rational eigenvalue problem (REP). Para-Hermitian rational matrices are square rational matrices that are Hermitian for all $z$ on the unit circle that are not poles. REPs are often solved via linearization, that is, using matrix pencils associated to the corresponding rational matrix that preserve the spectral structure. Yet, non-constant polynomial matrices cannot be para-Hermitian. Therefore, given a para-Hermitian rational matrix $R(z)$, we instead construct a $*$-palindromic linearization for $(1+z)R(z)$, whose eigenvalues that are not on the unit circle preserve the symmetries of the zeros and poles of $R(z)$. This task is achieved via Möbius transformations. We also give a constructive method that is based on an additive decomposition into the stable and anti-stable parts of $R(z)$. Analogous results are presented for para-skew-Hermitian rational matrices, i.e., rational matrices that are skew-Hermitian upon evaluation on those points of the unit circle that are not poles.

math.NA↗

arXiv2024

Efficient computation of Katz centrality for very dense networks via negative parameter Katz

Katz centrality (and its limiting case, eigenvector centrality) is a frequently used tool to measure the importance of a node in a network, and to rank the nodes accordingly. One reason for its popularity is that Katz centrality can be computed very efficiently when the network is sparse, i.e., having only $O(n)$ edges between its $n$ nodes. While sparsity is common in practice, in some applications one faces the opposite situation of a very dense network, where only $O(n)$ potential edges are missing with respect to a complete graph. We explain why and how, even for very dense networks, it is possible to efficiently compute the ranking stemming from Katz centrality for unweighted graphs, possibly directed and possibly with loops, by working on the complement graph. Our approach also provides an interpretation, regardless of sparsity, of "Katz centrality with negative parameter" as usual Katz centrality on the complement graph. For weighted graphs, we provide instead an approximation method that is based on removing sufficiently many edges from the network (or from its complement), and we give sufficient conditions for this approximation to provide the correct ranking. We include numerical experiments to illustrate the advantages of the proposed approach.

cs.SI↗