arXiv subjects

Ilya Spitkovsky

Publications and source records attributed to Ilya Spitkovsky.

11 recordsLinked to original sources

arXiv2025

The joint numerical range of three hermitian $4\times 4$ matrices

We analyze the joint numerical range $W$ of three hermitian matrices of order four. In the generic case, this three-dimensional convex set has a smooth boundary. We analyze non-generic structures. Fifteen possible classes regarding the numbers of non-elliptic faces in the boundary of $W$ are identified and an explicit example is presented for each class. Secondly, it is shown that a nonempty intersection of three mutually distinct one-dimensional faces is a corner point. Thirdly, introducing a tensor product structure into $\mathbb C^4=\mathbb C^2\otimes\mathbb C^2$, one defines the separable joint numerical range - a subset of $W$ useful in studies of quantum entanglement. The boundary of the separable numerical range is compared with that of $W$.

math.FA↗

arXiv2023

On the exact spectral factorization of rational matrix functions with applications to paraunitary filter banks

In this paper, we enhance a recent algorithm for approximate spectral factorization of matrix functions, extending its capabilities to precisely factorize rational matrices when an exact lower-upper triangular factorization is available. This novel approach leverages a fundamental component of the improved algorithm for the precise design of rational paraunitary filter banks, allowing for the predetermined placement of zeros and poles. The introduced algorithm not only advances the state-of-the-art in spectral factorization but also opens new avenues for the tailored design of paraunitary filters with specific spectral properties, offering significant potential for applications in signal processing and beyond.

math.CV↗

arXiv2023

Random Generator of Orthogonal Matrices in Finite Fields

We propose a superfast method for constructing orthogonal matrices $M\in\mathcal{O}(n,q)$ in finite fields $GF(q)$. It can be used to construct $n\times n$ orthogonal matrices in $Z_p$ with very high values of $n$ and $p$, and also orthogonal matrices with a certain circulant structure. Equally well one can construct paraunitary filter banks or wavelet matrices over finite fields. The construction mechanism is highly efficient, allowing for the complete screening and selection of an orthogonal matrix that meets specific constraints. For instance, one can generate a complete list of orthogonal matrices with given $n$ and $q=p^m$ provided that the order of $\mathcal{O}(n,q)$ is not too large. Although the method is based on randomness, isolated cases of failure can be identified well in advance of the basic procedure's start. The proposed procedures are based on the Janashia-Lagvilava method which was developed for an entirely different task, therefore, it may seem somewhat unexpected.

cs.IT↗

arXiv2021

An algorithm for J-spectral factorization of certain matrix functions

The problems of matrix spectral factorization and J-spectral factorization appear to be important for practical use in many MIMO control systems. We propose a numerical algorithm for J-spectral factorization which extends Janashia-Lagvilava matrix spectral factorization method to the indefinite case. The algorithm can be applied to matrices that have constant signatures for all leading principle submatrices. A numerical example is presented for illustrative purposes.

math.NA↗

arXiv2018

Inverse continuity of the numerical range map for Hilbert space operators

We describe continuity properties of the multivalued inverse of the numerical range map $f_A:x \mapsto \left\langle Ax, x \right\rangle$ associated with a linear operator $A$ defined on a complex Hilbert space $\mathcal{H}$. We prove in particular that $f_A^{-1}$ is strongly continuous at all points of the interior of the numerical range $W(A)$. We give examples where strong and weak continuity fail on the boundary and address special cases such as normal and compact operators.

math.FA↗

arXiv2018

Quantitative results on continuity of the spectral factorization mapping

The spectral factorization mapping $F\to F^+$ puts a positive definite integrable matrix function $F$ having an integrable logarithm of the determinant in correspondence with an outer analytic matrix function $F^+$ such that $F = F^+(F^+)^*$ almost everywhere. The main question addressed here is to what extent $\|F^+ - G^+\|_{H_2}$ is controlled by $\|F-G\|_{L_1}$ and $\|\log \det F - \log\det G\|_{L_1}$.

math.CV↗

arXiv2017

Robert Sheckley\s Answerer for two orthogonal projections

This paper is the written version of our talk (presented by the second author) at the IWOTA in Chemnitz in August 2017. The meta theorem of the paper is that Halmos' two projections theorem is something like Robert Sheckley's Answerer: no question about the W*- and C*-algebras generated by two orthogonal projections will go unanswered, provided the question is not foolish. An alternative approach to questions about two orthogonal projections makes use of the supersymmetry equality introduced by Avron, Seiler, and Simon. A noteworthy insight of the paper reveals that the supersymmetric approach is nothing but Halmos in different language and hence an equivalent Answerer.

math.FA↗

arXiv2017

Similarity between two projections

Given two orthogonal projections P and Q, we are interested in all unitary operators U such that UP=QU and UQ=PU. Such unitaries U have previously been constructed by Wang, Du, and Dou and also by one of the authors. One purpose of this note is to compare these constructions. Very recently, Dou, Shi, Cui, and Du described all unitaries U with the required property. Their proof is via the two projections theorem by Halmos. We here give a proof based on the supersymmetric approach by Avron, Seiler, and one of the authors.

math.FA↗

arXiv2016

Quantitative results on continuity of the spectral factorization mapping in the scalar case

In the scalar case, the spectral factorization mapping $f\to f^+$ puts a nonnegative integrable function $f$ having an integrable logarithm in correspondence with an outer analytic function $f^+$ such that $f = |f^+|^2$ almost everywhere. The main question addressed here is to what extent $\|f^+ - g^+\|_{H_2}$ is controlled by $\|f-g\|_{L_1}$ and $\|\log f - \log g\|_{L_1}$.

math.CV↗

arXiv2015

Inverse continuity on the boundary of the numerical range

Let $A \in M_n(\C)$. We consider the mapping $f_A(x)=x^*Ax$, defined on the unit sphere in $\C^n$. The map has a multi-valued inverse $f_A^{-1}$, and the continuity properties of $f_A^{-1}$ are considered in terms of the structure of the set of pre-images for points in the numerical range. It is shown that there may be only finitely many failures of continuity of $f_A^{-1}$, and conditions for where these failure occur are given. Additionally, we give a necessary and sufficient condition for weak inverse continuity to hold for $n=4$ and a sufficient condition for $n>4$.

math.FA↗

arXiv2013

Continuity properties of vectors realizing points in the classical field of values

For an $n$-by-$n$ matrix $A$, let $f_A$ be its "field of values generating function" defined as $f_A\colon x\mapsto x^*Ax$. We consider two natural versions of the continuity, which we call strong and weak, of $f_A^{-1}$ (which is of course multi-valued) on the field of values $F(A)$. The strong continuity holds, in particular, on the interior of $F(A)$, and at such points $z \in \partial F(A)$ which are either corner points, belong to the relative interior of flat portions of $\partial F(A)$, or whose preimage under $f_A$ is contained in a one-dimensional set. Consequently, $f_A^{-1}$ is continuous in this sense on the whole $F(A)$ for all normal, 2-by-2, and unitarily irreducible 3-by-3 matrices. Nevertheless, we show by example that the strong continuity of $f_A^{-1}$ fails at certain points of $\partial F(A)$ for some (unitarily reducible) 3-by-3 and (unitarily irreducible) 4-by-4 matrices. The weak continuity, in its turn, fails for some unitarily reducible 4-by-4 and untiarily irreducible 6-by-6 matrices.

math.FA↗