scholarly journals Explicit Block-Structures for Block-Symmetric Fiedler-like pencils

2018 ◽  
Vol 34 ◽  
pp. 472-499 ◽  
Author(s):  
M. I. Bueno ◽  
Madeline Martin ◽  
Javier Perez ◽  
Alexander Song ◽  
Irina Viviano
Keyword(s):  
Matrix Polynomial ◽  
Block Structure ◽  
Canonical Forms ◽  
Complex Matrix ◽  
Main Obstacle ◽  
Matrix Polynomials ◽  
Structured Matrix ◽  
Symmetric Structure ◽  
The Family ◽  
Block Structures

In the last decade, there has been a continued effort to produce families of strong linearizations of a matrix polynomial $P(\lambda)$, regular and singular, with good properties, such as, being companion forms, allowing the recovery of eigenvectors of a regular $P(\lambda)$ in an easy way, allowing the computation of the minimal indices of a singular $P(\lambda)$ in an easy way, etc. As a consequence of this research, families such as the family of Fiedler pencils, the family of generalized Fiedler pencils (GFP), the family of Fiedler pencils with repetition, and the family of generalized Fiedler pencils with repetition (GFPR) were constructed. In particular, one of the goals was to find in these families structured linearizations of structured matrix polynomials. For example, if a matrix polynomial $P(\lambda)$ is symmetric (Hermitian), it is convenient to use linearizations of $P(\lambda)$ that are also symmetric (Hermitian). Both the family of GFP and the family of GFPR contain block-symmetric linearizations of $P(\lambda)$, which are symmetric (Hermitian) when $P(\lambda)$ is. Now the objective is to determine which of those structured linearizations have the best numerical properties. The main obstacle for this study is the fact that these pencils are defined implicitly as products of so-called elementary matrices. Recent papers in the literature had as a goal to provide an explicit block-structure for the pencils belonging to the family of Fiedler pencils and any of its further generalizations to solve this problem. In particular, it was shown that all GFP and GFPR, after permuting some block-rows and block-columns, belong to the family of extended block Kronecker pencils, which are defined explicitly in terms of their block-structure. Unfortunately, those permutations that transform a GFP or a GFPR into an extended block Kronecker pencil do not preserve the block-symmetric structure. Thus, in this paper, the family of block-minimal bases pencils, which is closely related to the family of extended block Kronecker pencils, and whose pencils are also defined in terms of their block-structure, is considered as a source of canonical forms for block-symmetric pencils. More precisely, four families of block-symmetric pencils which, under some generic nonsingularity conditions are block minimal bases pencils and strong linearizations of a matrix polynomial, are presented. It is shown that the block-symmetric GFP and GFPR, after some row and column permutations, belong to the union of these four families. Furthermore, it is shown that, when $P(\lambda)$ is a complex matrix polynomial, any block-symmetric GFP and GFPR is permutationally congruent to a pencil in some of these four families. Hence, these four families of pencils provide an alternative but explicit approach to the block-symmetric Fiedler-like pencils existing in the literature.

scholarly journals The limit empirical spectral distribution of complex matrix polynomials

2021 ◽  
pp. 2250023
Author(s):  
Giovanni Barbarino ◽  
Vanni Noferini
Keyword(s):  
Spectral Distribution ◽  
Matrix Polynomial ◽  
Matrix Coefficient ◽  
Finite Variance ◽  
Complex Matrix ◽  
Hyperbolic Polynomials ◽  
Empirical Spectral Distribution ◽  
Matrix Polynomials ◽  
The Matrix ◽  
Uniformly Bounded

We study the empirical spectral distribution (ESD) for complex [Formula: see text] matrix polynomials of degree [Formula: see text] under relatively mild assumptions on the underlying distributions, thus highlighting universality phenomena. In particular, we assume that the entries of each matrix coefficient of the matrix polynomial have mean zero and finite variance, potentially allowing for distinct distributions for entries of distinct coefficients. We derive the almost sure limit of the ESD in two distinct scenarios: (1) [Formula: see text] with [Formula: see text] constant and (2) [Formula: see text] with [Formula: see text] bounded by [Formula: see text] for some [Formula: see text]; the second result additionally requires that the underlying distributions are continuous and uniformly bounded. Our results are universal in the sense that they depend on the choice of the variances and possibly on [Formula: see text] (if it is kept constant), but not on the underlying distributions. The results can be specialized to specific models by fixing the variances, thus obtaining matrix polynomial analogues of results known for special classes of scalar polynomials, such as Kac, Weyl, elliptic and hyperbolic polynomials.

A general inverse matrix series relation and associated polynomials – II

2021 ◽  
Vol 71 (2) ◽  
pp. 301-316
Author(s):  
Reshma Sanjhira
Keyword(s):  
Matrix Polynomial ◽  
Combinatorial Identities ◽  
Inverse Matrix ◽  
Matrix Polynomials ◽  
Special Cases ◽  
Series Representations ◽  
The Matrix ◽  
Associated Polynomials ◽  
Matrix Pair ◽  
Matrix Series

Abstract We propose a matrix analogue of a general inverse series relation with an objective to introduce the generalized Humbert matrix polynomial, Wilson matrix polynomial, and the Rach matrix polynomial together with their inverse series representations. The matrix polynomials of Kiney, Pincherle, Gegenbauer, Hahn, Meixner-Pollaczek etc. occur as the special cases. It is also shown that the general inverse matrix pair provides the extension to several inverse pairs due to John Riordan [An Introduction to Combinatorial Identities, Wiley, 1968].

scholarly journals A Schur-Cohn theorem for matrix polynomials

1990 ◽  
Vol 33 (3) ◽  
pp. 337-366 ◽  
Author(s):  
Harry Dym ◽  
Nicholas Young
Keyword(s):  
Unit Circle ◽  
Matrix Polynomial ◽  
Hermitian Matrix ◽  
Krein Space ◽  
Gram Matrix ◽  
Natural Basis ◽  
Test Matrix ◽  
Matrix Polynomials ◽  
Rational Vector ◽  
Square Matrix

Let N(λ) be a square matrix polynomial, and suppose det N is a polynomial of degree d. Subject to a certain non-singularity condition we construct a d by d Hermitian matrix whose signature determines the numbers of zeros of N inside and outside the unit circle. The result generalises a well known theorem of Schur and Cohn for scalar polynomials. The Hermitian “test matrix” is obtained as the inverse of the Gram matrix of a natural basis in a certain Krein space of rational vector functions associated with N. More complete results in a somewhat different formulation have been obtained by Lerer and Tismenetsky by other methods.

scholarly journals Stratigraphy of amethyst geode-bearing lavas and fault-block structures of the Entre Rios mining district, Paraná volcanic province, southern Brazil

2013 ◽  
Vol 86 (1) ◽  
pp. 187-198 ◽  
Author(s):  
LÉO A. HARTMANN ◽  
LUCAS M. ANTUNES ◽  
LEONARDO M. ROSENSTENGEL
Keyword(s):  
Block Structure ◽  
Underground Mines ◽  
Mining District ◽  
Volcanic Province ◽  
Fault Block ◽  
River Bed ◽  
World Class ◽  
Entre Rios ◽  
Block Structures

The Entre Rios mining district produces a large volume of amethyst geodes in underground mines and is part of the world class deposits in the Paraná volcanic province of South America. Two producing basalt flows are numbered 4 and 5 in the lava stratigraphy. A total of seven basalt flows and one rhyodacite flow are present in the district. At the base of the stratigraphy, beginning at the Chapecó river bed, two basalt flows are Esmeralda, low-Ti type. The third flow in the sequence is a rhyodacite, Chapecó type, Guarapuava subtype. Above the rhyodacite flow, four basalt flows are Pitanga, high-Ti type including the two mineralized flows; only the topmost basalt in the stratigraphy is a Paranapanema, intermediate-Ti type. Each individual flow is uniquely identified from its geochemical and gamma-spectrometric properties. The study of several sections in the district allowed for the identification of a fault-block structure. Blocks are elongated NW and the block on the west side of the fault was downthrown. This important structural characterization of the mining district will have significant consequences in the search for new amethyst geode deposits and in the understanding of the evolution of the Paraná volcanic province.

scholarly journals Linearization schemes for Hermite matrix polynomials

2015 ◽  
Author(s):  
Nikta Shayanfar ◽  
Heike Fassbender
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Matrix Polynomial ◽  
Multiple Input Multiple Output ◽  
Matrix Pencil ◽  
Least Square ◽  
Coupled Systems ◽  
Matrix Polynomials ◽  
Polynomial Eigenvalue Problem ◽  
The Matrix

The polynomial eigenvalue problem is to find the eigenpair of $(\lambda,x) \in \mathbb{C}\bigcup \{\infty\} \times \mathbb{C}^n \backslash \{0\}$ that satisfies $P(\lambda)x=0$, where $P(\lambda)=\sum_{i=0}^s P_i \lambda ^i$ is an $n\times n$ so-called matrix polynomial of degree $s$, where the coefficients $P_i, i=0,\cdots,s$, are $n\times n$ constant matrices, and $P_s$ is supposed to be nonzero. These eigenvalue problems arise from a variety of physical applications including acoustic structural coupled systems, fluid mechanics, multiple input multiple output systems in control theory, signal processing, and constrained least square problems. Most numerical approaches to solving such eigenvalue problems proceed by linearizing the matrix polynomial into a matrix pencil of larger size. Such methods convert the eigenvalue problem into a well-studied linear eigenvalue problem, and meanwhile, exploit and preserve the structure and properties of the original eigenvalue problem. The linearizations have been extensively studied with respect to the basis that the matrix polynomial is expressed in. If the matrix polynomial is expressed in a special basis, then it is desirable that its linearization be also expressed in the same basis. The reason is due to the fact that changing the given basis ought to be avoided \cite{H1}. The authors in \cite{ACL} have constructed linearization for different bases such as degree-graded ones (including monomial, Newton and Pochhammer basis), Bernstein and Lagrange basis. This contribution is concerned with polynomial eigenvalue problems in which the matrix polynomial is expressed in Hermite basis. In fact, Hermite basis is used for presenting matrix polynomials designed for matching a series of points and function derivatives at the prescribed nodes. In the literature, the linearizations of matrix polynomials of degree $s$, expressed in Hermite basis, consist of matrix pencils with $s+2$ blocks of size $n \times n$. In other words, additional eigenvalues at infinity had to be introduced, see e.g. \cite{CSAG}. In this research, we try to overcome this difficulty by reducing the size of linearization. The reduction scheme presented will gradually reduce the linearization to its minimal size making use of ideas from \cite{VMM1}. More precisely, for $n \times n$ matrix polynomials of degree $s$, we present linearizations of smaller size, consisting of $s+1$ and $s$ blocks of $n \times n$ matrices. The structure of the eigenvectors is also discussed.

Skewhermitian and mixed pencils

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Hermitian Matrix ◽  
The Other ◽  
Canonical Forms ◽  
Complex Field ◽  
Complex Matrix ◽  
Analogous Property ◽  
Matrix Pencils

This chapter turns to matrix pencils of the form A + tB, where one of the matrices A or B is skewhermitian and the other may be hermitian or skewhermitian. Canonical forms of such matrix pencils are given under strict equivalence and under simultaneous congruence, with full detailed proofs, again based on the Kronecker forms. Comparisons with real and complex matrix pencils are presented. In contrast to hermitian matrix pencils, two complex skewhermitian matrix pencils that are simultaneously congruent under quaternions need not be simultaneously congruent under the complex field, although an analogous property is valid for pencils of real skewsymmetric matrices. Similar results hold for real or complex matrix pencils A + tB, where A is real symmetric or complex hermitian and B is real skewsymmetric or complex skewhermitian.

scholarly journals Stability Analysis and Monitoring Method for the Key Block Structure of the Basic Roof of Noncoal Pillar Mining with Automatically Formed Gob-Side Entry

2019 ◽  
Vol 2019 ◽  
pp. 1-14 ◽  
Author(s):  
Jianning Liu ◽  
Manchao He ◽  
Yajun Wang ◽  
Ruifeng Huang ◽  
Jun Yang ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Coal Mine ◽  
Block Structure ◽  
Monitoring Method ◽  
Roof Caving ◽  
Pillar Mining ◽  
Key Block ◽  
The Stability ◽  
Block Structures ◽  
Basic Roof

The key block of the basic roof is the main contributor to the structural stability of a roadway. Research on the stability of the key block structure is of great significance for the promotion of noncoal pillar mining with automatically formed gob-side entry (GEFANM) technology. This paper is set in the engineering context of the GEFANM experiment at the Ningtiaota Coal Mine. The study fully considered the differences in the gob roof caving on the roof-cutting-line side, and the range of rotation angles to maintain a stable key block was determined. Based on this range of rotation angles, the range of safe bulking coefficients of gangue was calculated. The bulking coefficient of the gangue on the gravel side of the roadway was used as the metric in a new monitoring method and in the calculation of the field parameters. The range of safe bulking coefficients was determined to be 1.40–1.37. Field monitoring was conducted to obtain the gangue bulking coefficient on the gravel side. Combining the roof and floor convergence data, when the bulking coefficient fell within the safe range, the convergence was 95–113 mm. In this stage, the key block structure was stable. When the gangue bulking coefficient fell outside the safe range, the convergence was larger, and cracks were observed. The key block may be vulnerable to instability. The results affirmed that the gangue bulking coefficient can be used as a monitoring metric to study the stability of key block structures.

Learning Block Structures in U-statistic Based Matrices

Biometrika ◽  
2020 ◽  
Author(s):  
Weiping Zhang ◽  
Baisuo Jin ◽  
Zhidong Bai
Keyword(s):  
Matrix Theory ◽  
Block Structure ◽  
Group Structure ◽  
Asymptotic Properties ◽  
Finite Sample ◽  
Eigenvalues And Eigenvectors ◽  
Sample Matrix ◽  
U Statistics ◽  
Large Matrix ◽  
Block Structures

Abstract We introduce a conceptually simple, efficient and easily implemented approach for learning the block structure in a large matrix. Using the properties of U-statistics and large dimensional random matrix theory, the group structure of many variables can be directly identified based on the eigenvalues and eigenvectors of the scaled sample matrix. We also established the asymptotic properties of the proposed approach under mild conditions. The finite-sample performance of the approach is examined by extensive simulations and data examples.

scholarly journals Generalized Standard Triples for Algebraic Linearizations of Matrix Polynomials

2021 ◽  
Vol 37 ◽  
pp. 640-658
Author(s):  
Eunice Y.S. Chan ◽  
Robert M. Corless ◽  
Leili Rafiee Sevyeri
Keyword(s):  
Orthogonal Polynomial ◽  
Matrix Polynomial ◽  
Polynomial Basis ◽  
Regular Matrix ◽  
Bernstein Basis ◽  
Monomial Basis ◽  
Matrix Polynomials ◽  
Regular Matrix Polynomial ◽  
Polynomial Bases

We define generalized standard triples $\boldsymbol{X}$, $\boldsymbol{Y}$, and $L(z) = z\boldsymbol{C}_{1} - \boldsymbol{C}_{0}$, where $L(z)$ is a linearization of a regular matrix polynomial $\boldsymbol{P}(z) \in \mathbb{C}^{n \times n}[z]$, in order to use the representation $\boldsymbol{X}(z \boldsymbol{C}_{1}~-~\boldsymbol{C}_{0})^{-1}\boldsymbol{Y}~=~\boldsymbol{P}^{-1}(z)$ which holds except when $z$ is an eigenvalue of $\boldsymbol{P}$. This representation can be used in constructing so-called  algebraic linearizations for matrix polynomials of the form $\boldsymbol{H}(z) = z \boldsymbol{A}(z)\boldsymbol{B}(z) + \boldsymbol{C} \in \mathbb{C}^{n \times n}[z]$ from generalized standard triples of $\boldsymbol{A}(z)$ and $\boldsymbol{B}(z)$. This can be done even if $\boldsymbol{A}(z)$ and $\boldsymbol{B}(z)$ are expressed in differing polynomial bases. Our main theorem is that $\boldsymbol{X}$ can be expressed using the coefficients of the expression $1 = \sum_{k=0}^\ell e_k \phi_k(z)$ in terms of the relevant polynomial basis. For convenience, we tabulate generalized standard triples for orthogonal polynomial bases, the monomial basis, and Newton interpolational bases; for the Bernstein basis; for Lagrange interpolational bases; and for Hermite interpolational bases.

scholarly journals Vector Spaces of Generalized Linearizations for Rectangular Matrix Polynomials

2019 ◽  
Vol 35 ◽  
pp. 116-155
Author(s):  
Biswajit Das ◽  
Shreemayee Bora
Keyword(s):  
Eigenvalue Problem ◽  
Matrix Polynomial ◽  
Vector Spaces ◽  
Backward Error ◽  
Rectangular Matrix ◽  
Matrix Polynomials ◽  
Polynomial Eigenvalue Problem ◽  
Stable Manner ◽  
The Matrix ◽  
Matrix Pencils

The complete eigenvalue problem associated with a rectangular matrix polynomial is typically solved via the technique of linearization. This work introduces the concept of generalized linearizations of rectangular matrix polynomials. For a given rectangular matrix polynomial, it also proposes vector spaces of rectangular matrix pencils with the property that almost every pencil is a generalized linearization of the matrix polynomial which can then be used to solve the complete eigenvalue problem associated with the polynomial. The properties of these vector spaces are similar to those introduced in the literature for square matrix polynomials and in fact coincide with them when the matrix polynomial is square. Further, almost every pencil in these spaces can be `trimmed' to form many smaller pencils that are strong linearizations of the matrix polynomial which readily yield solutions of the complete eigenvalue problem for the polynomial. These linearizations are easier to construct and are often smaller than the Fiedler linearizations introduced in the literature for rectangular matrix polynomials. Additionally, a global backward error analysis applied to these linearizations shows that they provide a wide choice of linearizations with respect to which the complete polynomial eigenvalue problem can be solved in a globally backward stable manner.

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