scholarly journals Canonical forms for complex matrix congruence and ∗congruence

2006 ◽  
Vol 416 (2-3) ◽  
pp. 1010-1032 ◽  
Author(s):  
Roger A. Horn ◽  
Vladimir V. Sergeichuk
Keyword(s):  
Canonical Forms ◽  
Complex Matrix ◽  
Matrix Congruence

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 A note on canonical forms for matrix congruence

1996 ◽  
Vol 249 (1-3) ◽  
pp. 207-215 ◽  
Author(s):  
Jeffrey M. Lee ◽  
David A. Weinberg
Keyword(s):  
Canonical Forms ◽  
Matrix Congruence

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.

Appendix: Real and complex canonical forms

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Canonical Forms ◽  
Real Matrix ◽  
Complex Matrix ◽  
Complex Matrices ◽  
Matrix Pencils ◽  
Real Matrices

This chapter presents canonical forms for real and complex matrices and for pairs of real and complex matrices, or matrix pencils, with symmetries. All these forms are known, and most are well-known. The chapter first looks at Jordan and Kronecker canonical forms, before turning to real matrix pencils with symmetries. It provides canonical forms for pairs of real matrices, either one of which is symmetric or skewsymmetric, or what is the same, corresponding matrix pencils. Finally, this chapter presents canonical forms of complex matrix pencils with various symmetries, such as complex matrix pencils with symmetries with respect to transposition.

The Frechet Differential of a Primary Matrix Function

1973 ◽  
Vol 25 (3) ◽  
pp. 554-559 ◽  
Author(s):  
David L. Powers
Keyword(s):  
Power Series ◽  
Matrix Function ◽  
Canonical Forms ◽  
Complex Matrix ◽  
Fréchet Differential ◽  
Interpolating Polynomials

Let X be a given complex matrix of order n. If f(z) is analytic at the eigenvalues of X, one may define the primary matrix function f(X) with stem function f(z) by using any of several well-known methods: for instance, canonical forms, power series, or interpolating polynomials [9].

Seismite in the Devonian Sultan Formation, Frenchman Mountain, Nevada: Evidence of far-field effect of the Alamo Event

2016 ◽  
Vol 53 (2) ◽  
pp. 93-114
Author(s):  
Jesús Pinto ◽  
John Warme
Keyword(s):  
Carbonate Platform ◽  
Far Field ◽  
Carbonate Sediments ◽  
Impact Event ◽  
Complex Matrix ◽  
Near Surface ◽  
Fault Block ◽  
Southern Nevada ◽  
Salt Pan ◽  
The Alamo

We interpret a discrete, anomalous ~10-m-thick interval of the shallow-marine Middle to Late Devonian Valentine Member of the Sultan Formation at Frenchman Mountain, southern Nevada, to be a seismite, and that it was generated by the Alamo Impact Event. A suite of deformation structures characterize this unique interval of peritidal carbonate facies at the top of the Valentine Member; no other similar intervals have been discovered in the carbonate beds on Frenchman Mountain or in equivalent Devonian beds exposed in ranges of southern Nevada. The disrupted band extends for 5 km along the Mountain, and onto the adjoining Sunrise Mountain fault block for an additional 4+km. The interval displays a range of brittle, ductile and fluidized structures, and is divided into four informal bed-parallel units based on discrete deformation style and internal features that carry laterally across the study area. Their development is interpreted as the result of intrastratal compressional and contractional forces imposed upon the unconsolidated to fully cemented near-surface carbonate sediments at the top of the Valentine Member. The result is an assemblage of fractured, faulted, and brecciated beds, some of which were dilated, fluidized and injected to form new and complex matrix bands between beds. We interpret that the interval is an unusually thick and well displayed seismite. Because the Sultan Formation correlates northward to the Frasnian (lower Upper Devonian) carbonate rocks of the Guilmette Formation, and the Guilmette contains much thicker and more proximal exposures of the Alamo Impact Breccia, including seismites, we interpret the Frenchman Mountain seismite to be a far-field product of the Alamo Impact Event. Accompanying ground motion and deformation of the inner reaches of the Devonian carbonate platform may have resulted in a fall of relative sea level and abrupt shift to a salt-pan paleoenvironment exhibited by the post-event basal beds of the directly overlying Crystal Pass Member.

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