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Over a general field and for an arbitrary positive integer k, those triangular matrices whose kth power is diagonal are explicitly characterized. This is then used to characterize those complex mat...

We identify new classes of structured matrices whose numerical range is of the elliptical type, that is, an elliptical disk or the convex hull of elliptical disks.

UID/MAT/00297/2019 UID/MAT/04721/2019 We consider the class (Formula presented.) of symmetric (Formula presented.) -matrices with zero trace and constant row sums k which can be identified with the class of the adjacency matrices of k-regular undirected graphs. In a previous paper, two partial orders, the Bruhat and the Bruhat-graph order, have been introduced in this class. In fact, when k = 1 or k = 2, it was shown that the two orders coincide, while for (Formula presented.) the two orders are distinct. In this paper we give general properties of minimal and maximal matrices for these orders on (Formula presented.) and study the minimal and maximal matrices when k = 1, 2 or 3. publishersversion published

In this paper, a complete description of the linear maps $\phi:W_{n}\rightarrow W_{n}$ that preserve the Lorentz spectrum is given when $n=2$, and $W_{n}$ is the space $M_{n}$ of $n\times n$ real matrices or the subspace $S_{n}$ of $M_{n}$ formed by the symmetric matrices. In both cases, it has been shown that $\phi(A)=PAP^{-1}$ for all $A\in W_{2}$, where $P$ is a matrix with a certain structure. It was also shown that such preservers do not change the nature of the Lorentz eigenvalues (that is, the fact that they are associated with Lorentz eigenvectors in the interior or on the boundary of the Lorentz cone). These results extend to $n=2$ those for $n\geq 3$ obtained by Bueno, Furtado, and Sivakumar (2021). The case $n=2$ has some specificities, when compared to the case $n\geq3,$ due to the fact that the Lorentz cone in $\mathbb{R}^{2}$ is polyedral, contrary to what happens when it is contained in $\mathbb{R}^{n}$ with $n\geq3.$ Thus, the study of the Lorentz spectrum preservers on $W_n = M_n$ also follows from the known description of the Pareto spectrum preservers on $M_n$.

An n-by-n real symmetric matrix is called copositive if its quadratic form is nonnegative on nonnegative vectors. Our interest is in identifying which linear transformations on symmetric matrices preserve copositivity either in the into or onto sense. We conjecture that in the onto case, the map must be congruence by a monomial matrix (a permutation times a positive diagonal matrix). This is proven under each of some additional natural hypotheses. Also, the into preservers of standard type are characterized. A general characterization in the into case seems di¢ cult, and examples are given. One of them provides a counterexample to a conjecture about the into preservers., This work was partially supported by project UID/MAT/04721/2019 and by project UID/MAT/00013/2013

In this paper we consider sparse symmetrically banded matrices in which the nonzero off-diagonals are positioned a multiple of k steps from the main diagonal. We show that such a matrix T is permutationally similar to direct sum of banded matrices. In particular, when T has exactly one nonzero off-diagonal above and below the main diagonal, the direct summands are tridiagonal. If T has a w-Toeplitz structure, the blocks are w ′ -Toeplitz, with w ′ = lcm ( w , k ) / k . This reduction allows the study of spectral properties of T from those of the direct summands. Finally, we give a reduction of sparse symmetrically banded matrices relatively to the main antidiagonal.

In this article we study the distance d , measured in the Frobenius norm, of a tridiagonal matrix T to the set I T of similarly structured irreducible normal matrices. The matrices in the closure of I T whose distance to T is d are characterized. Known results in the literature for the cases in which T is real or a Toeplitz matrix are recovered. In addition, the special case in which T is a 2-Toeplitz matrix is considered.

In this paper we study the minimal matrices for the Bruhat order on the class of symmetric ( 0 , 1 ) -matrices with given row sum vector. We will show that, when restricted to the symmetric matrices, new minimal matrices may appear besides the symmetric matrices for the nonrestricted Bruhat order. We modify the algorithm presented by Brualdi and Hwang (2004), which gives a minimal matrix for the Bruhat order on the class of ( 0 , 1 ) -matrices with given row and column sum vectors, in order to obtain a minimal matrix for the Bruhat order on the class of symmetric ( 0 , 1 ) -matrices with given row sum vector. We identify other minimal matrices in some of these classes. Namely, we determine all the minimal matrices when the row sums are constant and equal to 3. We then describe a family of symmetric matrices that are minimal for the Bruhat order on the class of 2k-by-2k ( 0 , 1 ) -matrices with constant row sums equal to k + 1 and identify, in terms of the term rank of a matrix, a class of symmetric matrices that are related in the Bruhat order with one of these minimal matrices.

The computation of eigenvalues and eigenvectors of matrix polynomials is an important, but difficult, problem. The standard approach to solve this problem is to use linearizations, which are matrix polynomials of degree 1 that share the eigenvalues of P ( λ ) . Hermitian matrix polynomials and their real eigenvalues are of particular interest in applications. Attached to these eigenvalues is a set of signs called the sign characteristic. From both a theoretical and a practical point of view, it is important to be able to recover the sign characteristic of a Hermitian linearization of P ( λ ) from the sign characteristic of P ( λ ) . In this paper, for a Hermitian matrix polynomial P ( λ ) with nonsingular leading coefficient, we describe, in terms of the sign characteristic of P ( λ ) , the sign characteristic of the Hermitian linearizations in the vector space DL ( P ) (Mackey, Mackey, Mehl and Mehrmann, 2006). In particular, we identify the Hermitian linearizations in DL ( P ) that preserve the sign characteristic of P ( λ ) . We also provide a description of the sign characteristic of the Hermitian linearizations of P ( λ ) in the family of generalized Fiedler pencils with repetition (Bueno, Dopico, Furtado and Rychnovsky, 2015).

The standard way of solving numerically a polynomial eigenvalue problem (PEP) is to use a linearization and solve the corresponding generalized eigenvalue problem (GEP). In addition, if the PEP possesses one of the structures arising very often in applications, then the use of a linearization that preserves such structure combined with a structured algorithm for the GEP presents considerable numerical advantages. Block-symmetric linearizations have proven to be very useful for constructing structured linearizations of structured matrix polynomials. In this scenario, we analyze the eigenvalue condition numbers and backward errors of approximated eigenpairs of a block symmetric linearization that was introduced by Fiedler (Linear Algebra Appl 372:325–331, 2003) for scalar polynomials and generalized to matrix polynomials by Antoniou and Vologiannidis (Electron J Linear Algebra 11:78–87, 2004). This analysis reveals that such linearization has much better numerical properties than any other block-symmetric linearization analyzed so far in the literature, including those in the well known vector space $$\mathbb {DL}(P)$$ of block-symmetric linearizations. The main drawback of the analyzed linearization is that it can be constructed only for matrix polynomials of odd degree, but we believe that it will be possible to extend its use to even degree polynomials via some strategies in the near future.

Given a matrix polynomial P ( λ ) = ∑ i = 0 k λ i A i of degree k , where A i are n × n matrices with entries in a field F , the development of linearizations of P ( λ ) that preserve whatever structure P ( λ ) might posses has been a very active area of research in the last decade. Most of the structure-preserving linearizations of P ( λ ) discovered so far are based on certain modifications of block-symmetric linearizations. The block-symmetric linearizations of P ( λ ) available in the literature fall essentially into two classes: linearizations based on the so-called Fiedler pencils with repetition, which form a finite family, and a vector space of dimension k of block-symmetric pencils, called DL ( P ) , such that most of its pencils are linearizations. One drawback of the pencils in DL ( P ) is that none of them is a linearization when P ( λ ) is singular. In this paper we introduce new vector spaces of block-symmetric pencils, most of which are strong linearizations of P ( λ ) . The dimensions of these spaces are O ( n 2 ) , which, for n ≥ k , are much larger than the dimension of DL ( P ) . When k is odd, many of these vector spaces contain linearizations also when P ( λ ) is singular. The coefficients of the block-symmetric pencils in these new spaces can be easily constructed as k × k block-matrices whose n × n blocks are of the form 0, ± α I n , ± α A i , or arbitrary n × n matrices, where α is an arbitrary nonzero scalar.

In this paper we give strong linearizations of a matrix polynomial P ( λ ) preserving the skew-symmetry or T-alternating structure of P ( λ ) . The linearizations obtained are of the form S L ( λ ) , where L ( λ ) is a block-symmetric Fiedler pencil with repetition and S is a direct sum of blocks of the form I or − I , with I the identity matrix. This paper is a continuation of [2] , where the corresponding problem for P ( λ ) with a symmetric structure was studied and, as a consequence, the block-symmetric Fiedler pencils with repetition were characterized.

It is shown that for any k ∈ { 1 , 2 , … , ( m − 1 ) ! } there exist m invertible complex matrices such that among the m ! products A σ = A σ ( 1 ) A σ ( 2 ) ⋯ A σ ( m ) , σ ∈ S m , exactly k different similarity classes occur. The cases in which the matrices A i are upper triangular or are 2-by-2 are considered in detail. In the former case, it is shown that any m − 1 unispectral matrices with common eigenvalue may be found among the A σ , and in the latter case it is shown explicitly how to achieve ( m − 1 ) ! and ( m − 1 ) ! 2 similarity classes, as well as any number from 1 to 6 when m = 4 . Other particular results are given, as well as a discussion of further natural questions.

The development of strong linearizations preserving whatever structure a matrix polynomial might possess has been a very active area of research in the last years, since such linearizations are the starting point of numerical algorithms for computing eigenvalues of structured matrix polynomials with the properties imposed by the considered structure. In this context, Hermitian matrix polynomials are one of the most important classes of matrix polynomials arising in applications and their real eigenvalues are of great interest. The sign characteristic is a set of signs attached to these real eigenvalues which is crucial for determining the behavior of systems described by Hermitian matrix polynomials and, therefore, it is desirable to develop linearizations that preserve the sign characteristic of these polynomials, but, at present, only one such linearization is known. In this paper, we present a complete characterization of all the Hermitian strong linearizations that preserve the sign characteristic of a given Hermitian matrix polynomial and identify several families of such linearizations that can be constructed very easily from the coefficients of the polynomial.

We consider the problem of describing the possible exponents of [math] -by- [math] boolean primitive circulant matrices. It is well known that this set is a subset of [math] and not all integers in [math] are attainable exponents. In the literature, some attention has been paid to the gaps in the set of exponents. The first three gaps have been proven, that is, the integers in the intervals [math] , [math] and [math] are not attainable exponents. Here we study the distribution of exponents in between those gaps by giving the exact exponents attained there by primitive circulant matrices. We also study the distribution of exponents in between the third gap and our conjectured fourth gap. It is interesting to point out that the exponents attained in between the ( [math] )-th and the [math] -th gap depend on the value of [math] .

SUMMARY The main purpose of this paper is the extension of the classical spectral direct and inverse analysis of Jacobi matrices for the non-self-adjoint setting. Matrices of this class appear in the context of non-Hermitian quantum mechanics. The reconstruction of a pseudo-Jacobi matrix from its spectrum and the spectra of two complementary principal matrices is investigated in the context of indefinite inner product spaces. An existence and uniqueness theorem is given, and a strikingly simple algorithm, based on the Euclidean division algorithm, to reconstruct the matrix from the spectral data is presented. A result of Friedland and Melkman stating a necessary and sufficient condition for a real sequence to be the spectrum of a non-negative Jacobi matrix is revisited and generalized. Namely, it is shown that a suitable set of prescribed eigenvalues defines a unique non-negative pseudo-Jacobi matrix, which is J-Hermitian for a fixed J. Copyright © 2012 John Wiley & Sons, Ltd.

A quasi-Jacobi form for J -unitarily diagonalizable J -normal matrices is given, extending a result for normal matrices due to Malamud. The inverse eigenvalue problem for J -normal matrices satisfying certain prescribed spectral conditions is investigated. It is shown that there exists unicity in the case of pseudo-Jacobi matrices.

The problem of the existence of a J -normal matrix A when its spectrum and the spectrum of some of its ( n - 1 ) × ( n - 1 ) principal submatrices are prescribed is analyzed. The case of 3 × 3 matrices is particularly investigated. The results here obtained in the framework of indefinite inner product spaces are in the spirit of those due to Nylen, Tam and Uhlig.

The interlacing theorem of Cauchy–Poincare states that the eigenvalues of a principal submatrix A 0 of a Hermitian matrix A interlace the eigenvalues of A . Fan and Pall obtained an analog of this theorem for normal matrices. In this note we investigate analogs of Cauchy–Poincare and Fan–Pall interlacing theorems for J -Hermitian and J -normal matrices. The corresponding inverse spectral problems are also considered.

In earlier work, the possible spectra in a congruence class and the possible Jordan forms, among nilpotent matrices congruent to a given nilpotent matrix, were characterized. Here, we study the natural, but quite different, question of possible Jordan forms of matrices with a single nonzero eigenvalue (which may be taken to be 1) that are congruent to a given matrix with that eigenvalue. We focus upon a class of matrices that includes the essentially real matrices for which 0 is not in the field of values. Unlike the nilpotent case, the number of Jordan blocks, as well as their sizes, may change. Necessary conditions are derived and then it is shown that they are sufficient through a lengthy construction process. Some of the results go beyond the indicated case, but do not cover general matrices. §This work was done within the activities of Centro de Estruturas Lineares e Combinatorias da Universidade de Lisboa.

For each congruence class containing a nilpotent matrix, all possible nilpotent Jordan structures occurring in that class are determined. The only two constraints are: a fixed number of Jordan blocks, determined by the rank of the class, and a minimum number of one-by-one blocks, determined by the dimension of the intersection of the null spaces of A and A ∗ , for A representative of the class. An important tool, that may be of independent interest, is a unitary similarity version of the Jordan canonical form.

Two Hermitian matrices A,B∈Mn(C) are said to be Hermitian-congruent if there exists a nonsingular Hermitian matrix C∈Mn(C) such that B=CAC. In this paper, we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable Hermitian matrices A and B to be Hermitian-congruent. Moreover, when A and B are Hermitian-congruent, we describe the possible inertias of the Hermitian matrices C that carry the congruence. We also give necessary and sufficient conditions for any 2-by-2 nonsingular Hermitian matrices to be Hermitian-congruent. In both of the studied cases, we show that if A and B are real and Hermitian-congruent, then they are congruent by a real symmetric matrix. Finally we note that if A and B are 2-by-2 nonsingular real symmetric matrices having the same sign pattern, then there is always a real symmetric matrix C satisfying B=CAC. Moreover, if both matrices are positive, then C can be picked with arbitrary inertia.

Let A and B be n × n matrices over an algebraically closed field F. The pair ( A, B ) is said to be spectrally complete if, for every sequence c1,…,cn ∈F such that det (AB)=c1 ,…,cn , there exist matrices A′,B,′∈F,n×n similar to A, B, respectively, such that A′B′ has eigenvalues c1,…,cn . In this article, we describe the spectrally complete pairs. Assuming that A and B are nonsingular, the possible eigenvalues of A′B′ when A′ and B′ run over the sets of the matrices similar to A and B, respectively, were described in a previous article.

For A∈Mn(C), Aut(A), the congruential automorphism group of A, is the set of all non-singular C∈Mn(C) such that C∗AC=A. Here, the A for which Aut(A) is bounded are described, and this allows the characterization of pairs A, B for which the set of matrices carrying the congruence of A to B is compact. This characterization is in terms of non-singular unitoid matrices in which certain configurations of canonical angles are forbidden.

In this paper we describe the possible spectra among matrices congruent to a given nonsingular matrix A∈M n ( C ) such that 0 is a boundary point of its field of values. In the process we give a reducible form under congruence for A . If A is either singular or such that 0 is not a boundary point of its field of values, the possible spectra among matrices congruent to A were studied in [Linear and Multilinear Algebra 49 (2001) 243].

This paper studies the possibility of writing a given square matrix as the product of two matrices with prescribed spectra and ranks. It extends some previously known results.

Let A be a matrix with entries in the field of real numbers. In this paper we give necessary and sufficient conditions for the existence of real matrices B and C, with prescribed characteristic polynomials, such that A=BC.

In this paper we study the possible spectra among matrices congruent to a given A∊Mn (C). It is important to distinguish singular A from nonsingular and, among non-singular matrices, to distinguish...

We introduce the class of unitoid matrices as those that are diagonalizable by congruence; the nondegenerate canonical angles of a unitoid matrix A are the directions of the nonzero entries of a diagonal matrix congruent to A (which are shown to be unique). Sylvester's law states that two Hermitian matrices of the same size are congruent if and only if they have the same numbers of positive (respectively, negative) eigenvalues. Two unitoid matrices are congruent if and only if they have the same nondegenerate canonical angles.