Michael Tsatsomeros - Profile on Academia.edu (original) (raw)

Papers by Michael Tsatsomeros

Localization of the Spectrum of a Matrix

New and old results will be presented on the Envelope, E(A)E(A)E(A), which is a bounded region in the c... more New and old results will be presented on the Envelope, E(A)E(A)E(A), which is a bounded region in the complex plane that contains the eigenvalues of a complex matrix AAA. E(A)E(A)E(A) is the intersection of an infinite number of regions defined by elliptic curves. As such, E(A)E(A)E(A) resembles and is contained in the numerical range of AAA, which is the intersection of an infinite number of half-planes. The Envelope, however, can be much smaller than the numerical range, while not being much harder to compute. The talk is based on joint work with Panos Psarrakos, Maria Adam and Katerina Aretaki.

Q#-matrices and Q†-matrices: two extensions of the Q-matrix concept

Linear & Multilinear Algebra, Sep 25, 2021

A real square matrix A is called a Q-matrix if the linear complementarity problem LCP(A,q) has a ... more A real square matrix A is called a Q-matrix if the linear complementarity problem LCP(A,q) has a solution for all q∈Rn. This means that for every vector q, there exists a vector x≥0 such that y=Ax+...

Discrete Mathematics, Mar 1, 2018

As an inverse relation, involution with an invariant sequence plays a key role in combinatorics a... more As an inverse relation, involution with an invariant sequence plays a key role in combinatorics and features prominently in some of Shapiro's open questions [L.W. Shapiro, Some open questions about random walks, involutions, limiting distributions and generating functions, Adv. Appl. Math. 27 (2001) 585-596]. In this paper, invariant sequences are used to provide answers to some of these questions about the Fibonacci matrix and Riordan involutions.

Linear & Multilinear Algebra, Apr 27, 2017

Invertibility of all convex combinations of A and I is equivalent to the real eigenvalues of A, i... more Invertibility of all convex combinations of A and I is equivalent to the real eigenvalues of A, if any, being positive. Invertibility of all matrices whose rows are convex combinations of the respective rows of A and I is equivalent to A having positive principal minors (i.e., being a P-matrix). These results are extended by considering convex combinations of higher powers of A and of their rows. The invertibility of matrices in these convex hulls is associated with the eigenvalues of A lying in open sectors of the right-half plane and provides a general context for the theory of matrices with P-matrix powers.

arXiv (Cornell University), Jun 5, 2017

An infinite real sequence {a n } is called an invariant sequence of the first (resp., second) kin... more An infinite real sequence {a n } is called an invariant sequence of the first (resp., second) kind if We review and investigate invariant sequences of the first and second kinds, and study their relationships using similarities of Pascal-type matrices and their eigenspaces.

Electronic Journal of Linear Algebra, Feb 6, 2017

Eventually nonnegative matrices are square matrices whose powers become and remain (entrywise) no... more Eventually nonnegative matrices are square matrices whose powers become and remain (entrywise) nonnegative. Using classical Perron-Frobenius theory for cone preserving maps, this notion is generalized to matrices whose powers eventually leave a proper cone K ⊂ R n invariant, that is, A m K ⊆ K for all sufficiently large m. Also studied are the related notions of eventual cone invariance by the matrix exponential, as well as other generalizations of M-matrix and dynamical system notions.

Positivity, Jul 17, 2017

The semipositive cone of A ∈ R m×n , K A = {x ≥ 0 : Ax ≥ 0}, is considered mainly under the assum... more The semipositive cone of A ∈ R m×n , K A = {x ≥ 0 : Ax ≥ 0}, is considered mainly under the assumption that for some x ∈ K A , Ax > 0, namely, that A is a semipositive matrix. The duality of K A is studied and it is shown that K A is a proper polyhedral cone. The relation among semipositivity cones of two matrices is examined via generalized inverse positivity. Perturbations and intervals of semipositive matrices are discussed. Connections with certain matrix classes pertinent to linear complementarity theory are also studied.

Matrix Equalities and Inequalities

Chapman and Hall/CRC eBooks, Nov 2, 2006

Linear Algebra and its Applications, Oct 1, 2019

Semimonotone matrices A are those real matrices for which the operation Ax does not negate all po... more Semimonotone matrices A are those real matrices for which the operation Ax does not negate all positive entries of any nonzero, entrywise nonnegative vector x. In that respect, semimonotone matrices generalize the class of matrices all of whose principal minors are nonnegative. As such, they play an important role in the solution of the linear complementarity problem. Properties of semimonotone matrices that are largely analogous to the properties of the matrix classes they generalize are reviewed and developed.

Linear Algebra and its Applications, Jun 1, 2016

Semipositive matrices map a positive vector to a positive vector and as such they are a very broa... more Semipositive matrices map a positive vector to a positive vector and as such they are a very broad generalization of the irreducible nonnegative matrices. Nevertheless, the ensuing geometric mapping properties of semipositive matrices result in several parallels to the theory of cone preserving and cone mapping matrices. It is shown that for a semipositive matrix A, there exist a proper polyhedral cone K 1 of nonnegative vectors and a polyhedral cone K 2 of nonnegative vectors such that AK 1 = K 2 . The set of all nonnegative vectors mapped by A to the nonnegative orthant is a proper polyhedral cone; as a consequence, A belongs to a proper polyhedral cone comprising semipositive matrices. When the powers A k (k = 0, 1, . . .) have a common semipositivity vector, then A has a positive eigenvalue. If A has a sole peripheral eigenvalue λ and the powers of A have a common semipositivity vector with a non-vanishing term in the direction of the left eigenspace of λ, then A leaves a proper cone invariant.

Electronic Journal of Linear Algebra, 2009

An M ∨ -matrix has the form A = sI -B, where s ≥ ρ(B) ≥ 0 and B is eventually nonnegative; i.e., ... more An M ∨ -matrix has the form A = sI -B, where s ≥ ρ(B) ≥ 0 and B is eventually nonnegative; i.e., B k is entrywise nonnegative for all sufficiently large integers k. A theory of M ∨ -matrices is developed here that parallels the theory of M-matrices, in particular as it regards exponential nonnegativity, spectral properties, semipositivity, monotonicity, inverse nonnegativity and diagonal dominance.

arXiv (Cornell University), Jan 19, 2020

N-matrices are real n × n matrices all of whose principal minors are negative. We provide (i) an ... more N-matrices are real n × n matrices all of whose principal minors are negative. We provide (i) an O(2 n ) test to detect whether or not a given matrix is an N-matrix, and (ii) a characterization of N-matrices, leading to the recursive construction of every N-matrix.

All of Whose Powers Are Toeplitz

Let a, b and c be fixed complex numbers. Let Mn(a, b, c) be the n× n Toeplitz matrix all of whose... more Let a, b and c be fixed complex numbers. Let Mn(a, b, c) be the n× n Toeplitz matrix all of whose entries above the diagonal are a, all of whose entries below the diagonal are b, and all of whose entries on the diagonal are c. For 16 k6 n, each k× k principal minor of Mn(a, b, c) has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of Mn(a, b, c). We also show that all complex polynomials in Mn(a, b, c) are Toeplitz matrices. In particular, the inverse of Mn(a, b, c) is a Toeplitz matrix when it exists.

This encyclopedic book gives a detailed overview of our current (2019) knowledge of positive matr... more This encyclopedic book gives a detailed overview of our current (2019) knowledge of positive matrices and related concepts in the theory of matrix positivity. Its contents cover 208 pages and an extensive bibliography of 16 pages counts over 320 items from 1912 to 2019. The incipit of all modern studies in this field is the Perron-Frobenius theorem (1907)(1908). The book starts with a list of 53 subject specific symbols and a list of acronyms for 40 matrix classes. Chapter 1 briefly states a few mathematical concepts such as matrices, vectors, convexity, Helly's theorem, half-spaces and cones. Chapter 2 defines eleven specific positive-like matrix classes with some of their subclasses and acronyms and it states a complete containment diagram for ten of these. Chapters 3-6 deal with four of these classes with more details. These chapters develop our knowledge about semipositive matrices, P-matrices, inverse M-matrices, and copositive matrices with many newly developed short and long proofs. Chapter 3 alone contains 45 proofs of theorems and corollaries in 40 pages. Chapters 4 and 5 present interesting open research questions. Section 4.6 contains a recursive algorithm for the P-problem and Section 4.7.2 an algorithm for constructing P-matrices. Throughout the book there are applications of various positive-like matrices, such the linear complementarity problem (in Chapter 4) and iterative matrix algorithms and differential equation solvers (Chapter 5). The style of writing is very concise and full of minutiae in every page. Highlights are not stressed. Almost a quarter of the references refers to works authored or coauthored by the three authors of this book. This is a very personal assessment and a beautiful representation and re-interpretation of much of the authors' extensive work on this subject. However, standard generalizations, such as generalized Cholesky methods for symmetric matrices, inertia counting, polar decompositions and product representations by positive matrices, are missing. Neither sums of positive matrices, nor matrix pencil characterizations via matrix positivity are included. The book is almost typo-free, except of a notational inconsistency for generalized binomial coefficients that appear both in (correct) braces and in bold brackets (see pages 101-102). This is an excellent book for mathematicians, written by three experts in the theoretical aspects of this field. It could and should be an important resource for researchers in numerical analysis and nonmathematicians interested in computational and applied areas where positive-like matrices are encountered. It would be worth to have an appendix where the authors mention in which fields positivity matrix classes can occur. Reviewer: Frank Uhlig (Auburn) 15-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra 15B48 Positive matrices and their generalizations; cones of matrices Cited in 1 Review Cited in 9 Documents

Electronic Journal of Linear Algebra, 2012

The n × n Brualdi-Li matrix Bn has recently been shown to have maximal Perron value (spectral rad... more The n × n Brualdi-Li matrix Bn has recently been shown to have maximal Perron value (spectral radius) ρ among all tournament matrices of even order n, thus settling the conjecture by the same name. This renews our interest in estimating ρ and motivates us to study the Perron eigenvector x of Bn, which is normalized to have 1-norm equal to one. It follows that x minimizes the 2-norm among all Perron vectors of n × n tournament matrices. There are also interesting relations among the entries of x and ρ, allowing us to rank the teams corresponding to a Brualdi-Li tournament according to the Kendall-Wei and Ramanajucharyula ranking schemes.

Special Matrices, 2021

Continuous-time Markov chains have transition matrices that vary continuously in time. Classical ... more Continuous-time Markov chains have transition matrices that vary continuously in time. Classical theory of nonnegative matrices, M-matrices and matrix exponentials is used in the literature to study their dynamics, probability distributions and other stochastic properties. For the bene t of Perron-Frobenius cognoscentes, this theory is surveyed and further adapted to study continuous-time Markov chains on nite state spaces.

The Electronic Journal of Linear Algebra

In this article, some new results on MMM-matrices, HHH-matrices and their inverse classes are pro... more In this article, some new results on MMM-matrices, HHH-matrices and their inverse classes are proved. Specifically, we study when a singular ZZZ-matrix is an MMM-matrix, convex combinations of HHH-matrices, almost monotone HHH-matrices and Cholesky factorizations of HHH-matrices.

Localization of the Spectrum of a Matrix

New and old results will be presented on the Envelope, E(A)E(A)E(A), which is a bounded region in the c... more New and old results will be presented on the Envelope, E(A)E(A)E(A), which is a bounded region in the complex plane that contains the eigenvalues of a complex matrix AAA. E(A)E(A)E(A) is the intersection of an infinite number of regions defined by elliptic curves. As such, E(A)E(A)E(A) resembles and is contained in the numerical range of AAA, which is the intersection of an infinite number of half-planes. The Envelope, however, can be much smaller than the numerical range, while not being much harder to compute. The talk is based on joint work with Panos Psarrakos, Maria Adam and Katerina Aretaki.

Book reviewMatrices: Algebra, Analysis and Applications, Shmuel Friedland, World Scientific Publishing Co., New Jersey (2016), ISBN: 978-9814667968

Linear Algebra and its Applications, 2016

Recursive rank one perturbations for pole placement and cone reachability

Applied Mathematics and Computation, 2022

Localization of the Spectrum of a Matrix

New and old results will be presented on the Envelope, E(A)E(A)E(A), which is a bounded region in the c... more New and old results will be presented on the Envelope, E(A)E(A)E(A), which is a bounded region in the complex plane that contains the eigenvalues of a complex matrix AAA. E(A)E(A)E(A) is the intersection of an infinite number of regions defined by elliptic curves. As such, E(A)E(A)E(A) resembles and is contained in the numerical range of AAA, which is the intersection of an infinite number of half-planes. The Envelope, however, can be much smaller than the numerical range, while not being much harder to compute. The talk is based on joint work with Panos Psarrakos, Maria Adam and Katerina Aretaki.

Q#-matrices and Q†-matrices: two extensions of the Q-matrix concept

Linear & Multilinear Algebra, Sep 25, 2021

A real square matrix A is called a Q-matrix if the linear complementarity problem LCP(A,q) has a ... more A real square matrix A is called a Q-matrix if the linear complementarity problem LCP(A,q) has a solution for all q∈Rn. This means that for every vector q, there exists a vector x≥0 such that y=Ax+...

Discrete Mathematics, Mar 1, 2018

As an inverse relation, involution with an invariant sequence plays a key role in combinatorics a... more As an inverse relation, involution with an invariant sequence plays a key role in combinatorics and features prominently in some of Shapiro's open questions [L.W. Shapiro, Some open questions about random walks, involutions, limiting distributions and generating functions, Adv. Appl. Math. 27 (2001) 585-596]. In this paper, invariant sequences are used to provide answers to some of these questions about the Fibonacci matrix and Riordan involutions.

Linear & Multilinear Algebra, Apr 27, 2017

Invertibility of all convex combinations of A and I is equivalent to the real eigenvalues of A, i... more Invertibility of all convex combinations of A and I is equivalent to the real eigenvalues of A, if any, being positive. Invertibility of all matrices whose rows are convex combinations of the respective rows of A and I is equivalent to A having positive principal minors (i.e., being a P-matrix). These results are extended by considering convex combinations of higher powers of A and of their rows. The invertibility of matrices in these convex hulls is associated with the eigenvalues of A lying in open sectors of the right-half plane and provides a general context for the theory of matrices with P-matrix powers.

arXiv (Cornell University), Jun 5, 2017

An infinite real sequence {a n } is called an invariant sequence of the first (resp., second) kin... more An infinite real sequence {a n } is called an invariant sequence of the first (resp., second) kind if We review and investigate invariant sequences of the first and second kinds, and study their relationships using similarities of Pascal-type matrices and their eigenspaces.

Electronic Journal of Linear Algebra, Feb 6, 2017

Eventually nonnegative matrices are square matrices whose powers become and remain (entrywise) no... more Eventually nonnegative matrices are square matrices whose powers become and remain (entrywise) nonnegative. Using classical Perron-Frobenius theory for cone preserving maps, this notion is generalized to matrices whose powers eventually leave a proper cone K ⊂ R n invariant, that is, A m K ⊆ K for all sufficiently large m. Also studied are the related notions of eventual cone invariance by the matrix exponential, as well as other generalizations of M-matrix and dynamical system notions.

Positivity, Jul 17, 2017

The semipositive cone of A ∈ R m×n , K A = {x ≥ 0 : Ax ≥ 0}, is considered mainly under the assum... more The semipositive cone of A ∈ R m×n , K A = {x ≥ 0 : Ax ≥ 0}, is considered mainly under the assumption that for some x ∈ K A , Ax > 0, namely, that A is a semipositive matrix. The duality of K A is studied and it is shown that K A is a proper polyhedral cone. The relation among semipositivity cones of two matrices is examined via generalized inverse positivity. Perturbations and intervals of semipositive matrices are discussed. Connections with certain matrix classes pertinent to linear complementarity theory are also studied.

Matrix Equalities and Inequalities

Chapman and Hall/CRC eBooks, Nov 2, 2006

Linear Algebra and its Applications, Oct 1, 2019

Semimonotone matrices A are those real matrices for which the operation Ax does not negate all po... more Semimonotone matrices A are those real matrices for which the operation Ax does not negate all positive entries of any nonzero, entrywise nonnegative vector x. In that respect, semimonotone matrices generalize the class of matrices all of whose principal minors are nonnegative. As such, they play an important role in the solution of the linear complementarity problem. Properties of semimonotone matrices that are largely analogous to the properties of the matrix classes they generalize are reviewed and developed.

Linear Algebra and its Applications, Jun 1, 2016

Semipositive matrices map a positive vector to a positive vector and as such they are a very broa... more Semipositive matrices map a positive vector to a positive vector and as such they are a very broad generalization of the irreducible nonnegative matrices. Nevertheless, the ensuing geometric mapping properties of semipositive matrices result in several parallels to the theory of cone preserving and cone mapping matrices. It is shown that for a semipositive matrix A, there exist a proper polyhedral cone K 1 of nonnegative vectors and a polyhedral cone K 2 of nonnegative vectors such that AK 1 = K 2 . The set of all nonnegative vectors mapped by A to the nonnegative orthant is a proper polyhedral cone; as a consequence, A belongs to a proper polyhedral cone comprising semipositive matrices. When the powers A k (k = 0, 1, . . .) have a common semipositivity vector, then A has a positive eigenvalue. If A has a sole peripheral eigenvalue λ and the powers of A have a common semipositivity vector with a non-vanishing term in the direction of the left eigenspace of λ, then A leaves a proper cone invariant.

Electronic Journal of Linear Algebra, 2009

An M ∨ -matrix has the form A = sI -B, where s ≥ ρ(B) ≥ 0 and B is eventually nonnegative; i.e., ... more An M ∨ -matrix has the form A = sI -B, where s ≥ ρ(B) ≥ 0 and B is eventually nonnegative; i.e., B k is entrywise nonnegative for all sufficiently large integers k. A theory of M ∨ -matrices is developed here that parallels the theory of M-matrices, in particular as it regards exponential nonnegativity, spectral properties, semipositivity, monotonicity, inverse nonnegativity and diagonal dominance.

arXiv (Cornell University), Jan 19, 2020

N-matrices are real n × n matrices all of whose principal minors are negative. We provide (i) an ... more N-matrices are real n × n matrices all of whose principal minors are negative. We provide (i) an O(2 n ) test to detect whether or not a given matrix is an N-matrix, and (ii) a characterization of N-matrices, leading to the recursive construction of every N-matrix.

All of Whose Powers Are Toeplitz

Let a, b and c be fixed complex numbers. Let Mn(a, b, c) be the n× n Toeplitz matrix all of whose... more Let a, b and c be fixed complex numbers. Let Mn(a, b, c) be the n× n Toeplitz matrix all of whose entries above the diagonal are a, all of whose entries below the diagonal are b, and all of whose entries on the diagonal are c. For 16 k6 n, each k× k principal minor of Mn(a, b, c) has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of Mn(a, b, c). We also show that all complex polynomials in Mn(a, b, c) are Toeplitz matrices. In particular, the inverse of Mn(a, b, c) is a Toeplitz matrix when it exists.

This encyclopedic book gives a detailed overview of our current (2019) knowledge of positive matr... more This encyclopedic book gives a detailed overview of our current (2019) knowledge of positive matrices and related concepts in the theory of matrix positivity. Its contents cover 208 pages and an extensive bibliography of 16 pages counts over 320 items from 1912 to 2019. The incipit of all modern studies in this field is the Perron-Frobenius theorem (1907)(1908). The book starts with a list of 53 subject specific symbols and a list of acronyms for 40 matrix classes. Chapter 1 briefly states a few mathematical concepts such as matrices, vectors, convexity, Helly's theorem, half-spaces and cones. Chapter 2 defines eleven specific positive-like matrix classes with some of their subclasses and acronyms and it states a complete containment diagram for ten of these. Chapters 3-6 deal with four of these classes with more details. These chapters develop our knowledge about semipositive matrices, P-matrices, inverse M-matrices, and copositive matrices with many newly developed short and long proofs. Chapter 3 alone contains 45 proofs of theorems and corollaries in 40 pages. Chapters 4 and 5 present interesting open research questions. Section 4.6 contains a recursive algorithm for the P-problem and Section 4.7.2 an algorithm for constructing P-matrices. Throughout the book there are applications of various positive-like matrices, such the linear complementarity problem (in Chapter 4) and iterative matrix algorithms and differential equation solvers (Chapter 5). The style of writing is very concise and full of minutiae in every page. Highlights are not stressed. Almost a quarter of the references refers to works authored or coauthored by the three authors of this book. This is a very personal assessment and a beautiful representation and re-interpretation of much of the authors' extensive work on this subject. However, standard generalizations, such as generalized Cholesky methods for symmetric matrices, inertia counting, polar decompositions and product representations by positive matrices, are missing. Neither sums of positive matrices, nor matrix pencil characterizations via matrix positivity are included. The book is almost typo-free, except of a notational inconsistency for generalized binomial coefficients that appear both in (correct) braces and in bold brackets (see pages 101-102). This is an excellent book for mathematicians, written by three experts in the theoretical aspects of this field. It could and should be an important resource for researchers in numerical analysis and nonmathematicians interested in computational and applied areas where positive-like matrices are encountered. It would be worth to have an appendix where the authors mention in which fields positivity matrix classes can occur. Reviewer: Frank Uhlig (Auburn) 15-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to linear algebra 15B48 Positive matrices and their generalizations; cones of matrices Cited in 1 Review Cited in 9 Documents

Electronic Journal of Linear Algebra, 2012

The n × n Brualdi-Li matrix Bn has recently been shown to have maximal Perron value (spectral rad... more The n × n Brualdi-Li matrix Bn has recently been shown to have maximal Perron value (spectral radius) ρ among all tournament matrices of even order n, thus settling the conjecture by the same name. This renews our interest in estimating ρ and motivates us to study the Perron eigenvector x of Bn, which is normalized to have 1-norm equal to one. It follows that x minimizes the 2-norm among all Perron vectors of n × n tournament matrices. There are also interesting relations among the entries of x and ρ, allowing us to rank the teams corresponding to a Brualdi-Li tournament according to the Kendall-Wei and Ramanajucharyula ranking schemes.

Special Matrices, 2021

Continuous-time Markov chains have transition matrices that vary continuously in time. Classical ... more Continuous-time Markov chains have transition matrices that vary continuously in time. Classical theory of nonnegative matrices, M-matrices and matrix exponentials is used in the literature to study their dynamics, probability distributions and other stochastic properties. For the bene t of Perron-Frobenius cognoscentes, this theory is surveyed and further adapted to study continuous-time Markov chains on nite state spaces.

The Electronic Journal of Linear Algebra

In this article, some new results on MMM-matrices, HHH-matrices and their inverse classes are pro... more In this article, some new results on MMM-matrices, HHH-matrices and their inverse classes are proved. Specifically, we study when a singular ZZZ-matrix is an MMM-matrix, convex combinations of HHH-matrices, almost monotone HHH-matrices and Cholesky factorizations of HHH-matrices.

Localization of the Spectrum of a Matrix

New and old results will be presented on the Envelope, E(A)E(A)E(A), which is a bounded region in the c... more New and old results will be presented on the Envelope, E(A)E(A)E(A), which is a bounded region in the complex plane that contains the eigenvalues of a complex matrix AAA. E(A)E(A)E(A) is the intersection of an infinite number of regions defined by elliptic curves. As such, E(A)E(A)E(A) resembles and is contained in the numerical range of AAA, which is the intersection of an infinite number of half-planes. The Envelope, however, can be much smaller than the numerical range, while not being much harder to compute. The talk is based on joint work with Panos Psarrakos, Maria Adam and Katerina Aretaki.

Book reviewMatrices: Algebra, Analysis and Applications, Shmuel Friedland, World Scientific Publishing Co., New Jersey (2016), ISBN: 978-9814667968

Linear Algebra and its Applications, 2016

Recursive rank one perturbations for pole placement and cone reachability

Applied Mathematics and Computation, 2022