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.

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 The determinant of a complex matrix and Gershgorin circles

2019 ◽  
Vol 35 ◽  
pp. 181-186 ◽  
Author(s):  
Florian Bünger ◽  
Siegfried Rump
Keyword(s):  
Linear Algebra ◽  
Connected Component ◽  
Complex Matrix ◽  
Outer Loop ◽  
Complex Matrices ◽  
Real Matrices

Each connected component of the Gershgorin circles of a matrix contains exactly as many eigenvalues as circles are involved. Thus, the Minkowski (set) product of all circles contains the determinant if all circles are disjoint. In [S.M. Rump. Bounds for the determinant by Gershgorin circles. Linear Algebra and its Applications, 563:215--219, 2019.], it was proved that statement to be true for real matrices whose circles need not to be disjoint. Moreover, it was asked whether the statement remains true for complex matrices. This note answers that in the affirmative. As a by-product, a parameterization of the outer loop of a Cartesian oval without case distinction is derived.

scholarly journals Rational Criteria for Diagonalizability of Real Matrices

2020 ◽  
Vol 36 (36) ◽  
pp. 664-677
Author(s):  
João Ferreira Alves
Keyword(s):  
Linear Algebra ◽  
Real Matrix ◽  
Complex Matrices ◽  
Gram Matrices ◽  
The Matrix ◽  
The Moment ◽  
Real Matrices

The purpose of this note is to obtain rational criteria for diagonalizability of real matrices through the analysis of the moment and Gram matrices associated to a given real matrix. These concepts were introduced by Horn and Lopatin in [R.A. Horn and A.K. Lopatin. The moment and Gram matrices, distinct eigenvalues and zeroes, and rational criteria for diagonalizability. Linear Algebra and its Applications, 299:153-163, 1999] for complex matrices. However, when the matrix is real, it is possible to combine their results with the Borchardt-Jacobi Theorem, in order to get new and noteworthy rational criteria.

VI.—On Singular Pencils of Zehfuss, Compound, and Schläflian Matrices

1937 ◽  
Vol 56 ◽  
pp. 50-89 ◽  
Author(s):  
W. Ledermann
Keyword(s):  
Canonical Form ◽  
Direct Product ◽  
Canonical Forms ◽  
Singular Case ◽  
Elementary Divisors ◽  
Compound Matrices ◽  
Matrix Pencils ◽  
The Subject

In this paper the canonical form of matrix pencils will be discussed which are based on a pair of direct product matrices (Zehfuss matrices), compound matrices, or Schläflian matrices derived from given pencils whose canonical forms are known.When all pencils concerned are non-singular (i.e. when their determinants do not vanish identically), the problem is equivalent to finding the elementary divisors of the pencil. This has been solved by Aitken (1935), Littlewood (1935), and Roth (1934). In the singular case, however, the so-called minimal indices or Kronecker Invariants have to be determined in addition to the elementary divisors (Turnbull and Aitken, 1932, chap. ix). The solution of this problem is the subject of the following investigation.

Eigenvalue localization for complex matrices

2014 ◽  
Vol 27 ◽  
Author(s):  
Ibrahim Gumus ◽  
Omar Hirzallah ◽  
Fuad Kittaneh
Keyword(s):  
Frobenius Norm ◽  
Complex Matrix ◽  
Complex Matrices ◽  
Eigenvalue Localization ◽  
The Matrix

Let $A$ be an $n\times n$ complex matrix with $n\geq 3$. It is shown that at least $n-2$ of the eigenvalues of $A$ lie in the disk \begin{equation*}\left\vert z-\frac{\func{tr}A}{n}\right\vert \leq \sqrt{\frac{n-1}{n}\left(\sqrt{\left( \left\Vert A\right\Vert _{2}^{2}-\frac{\left\vert \func{tr} A\right\vert ^{2}}{n}\right) ^{2}-\frac{\left\Vert A^{\ast }A-AA^{\ast}\right\Vert _{2}^{2}}{2}}-\frac{\limfunc{spd}\nolimits^{2}(A)}{2}\right) },\end{equation*} where $\left\Vert A\right\Vert _{2},$ $\func{tr}A$, and $\limfunc{spd}(A)$ denote the Frobenius norm, the trace, and the spread of $A$, respectively. In particular, if $A=\left[ a_{ij}\right] $ is normal, then at least $n-2$ of the eigenvalues of $A$ lie in the disk {\small \begin{eqnarray*} & & \left\vert z-\frac{\func{tr}A}{n}\right\vert \\ & & \leq \sqrt{\frac{n-1}{n}\left( \frac{\left\Vert A\right\Vert _{2}^{2}}{2}-\frac{\left\vert \func{tr}A\right\vert ^{2}}{n}-\frac{3}{2}\max_{i,j=1,\dots,n} \left( \sum_{\substack{ k=1 \\ k\neq i}}^{n}\left\vert a_{ki}\right\vert ^{2}+\sum_{\substack{ k=1 \\ k\neq j}}^{n}\left\vert a_{kj}\right\vert ^{2}+\frac{\left\vert a_{ii}-a_{jj}\right\vert ^{2}}{2}\right) \right) }. \end{eqnarray*}} Moreover, the constant $\frac{3}{2}$ can be replaced by $4$ if the matrix $A$ is Hermitian.

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 Pólya's Permanent Problem

10.37236/1832 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
William McCuaig
Keyword(s):  
Polynomial Algorithms ◽  
Real Matrix ◽  
Nonsingular Matrices ◽  
Structural Characterizations ◽  
Real Matrices

A square real matrix is sign-nonsingular if it is forced to be nonsingular by its pattern of zero, negative, and positive entries. We give structural characterizations of sign-nonsingular matrices, digraphs with no even length dicycles, and square nonnegative real matrices whose permanent and determinant are equal. The structural characterizations, which are topological in nature, imply polynomial algorithms.

scholarly journals Computing The Permanent of (Some) Complex Matrices

2021 ◽  
pp. 753-761
Author(s):  
Pooja Mishra
Keyword(s):  
Relative Error ◽  
Deterministic Algorithm ◽  
Complex Matrix ◽  
Complex Matrices

We present a deterministic algorithm, which, for any given 0 < s <1 and an n × n real or complex matrix A = (aij) such that |aij − 1| ≤ 0.19 for all i, j computes the permanent of A within relative error s in nO(lnn−lns) time. The method can be extended to computing hafnians and multidimensional permanents.

scholarly journals On the cardinality of complex matrix scalings

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
George Hutchinson
Keyword(s):  
Upper Bound ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Complex Matrix ◽  
Real Matrices

AbstractWe disprove a conjecture made by Rajesh Pereira and Joanna Boneng regarding the upper bound on the number of doubly quasi-stochastic scalings of an n × n positive definite matrix. In doing so, we arrive at the true upper bound for 3 × 3 real matrices, and demonstrate that there is no such bound when n ≥ 4.

A Unitary Relation Between a Matrix and its Transpose

1970 ◽  
Vol 13 (2) ◽  
pp. 279-280 ◽  
Author(s):  
William R. Gordon
Keyword(s):  
Simple Relation ◽  
Complex Matrix ◽  
Complex Matrices

It is well known that if A is an n × n complex matrix and AT is its transpose, then there is an invertible n x n complex matrix S such that AT = S-1AS. In this note we wish to point out another simple relation between A and AT.If A is an n × n complex matrix and AT is its transpose then there are unitary n × n complex matrices U and V such that AT = UAV.

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