scholarly journals Sign Characteristics of Regular Hermitian Matrix Pencils under Generic Rank-1 and Rank-2 Perturbations

2015 ◽  
Vol 30 ◽  
pp. 760-794 ◽  
Author(s):  
Leonhard Batzke
Keyword(s):  
Canonical Form ◽  
Hermitian Matrix ◽  
Structure Preserving ◽  
Spectral Behavior ◽  
Generic Rank ◽  
Jordan Structure ◽  
Matrix Pencils ◽  
Rank 2 ◽  
Infinite Blocks

The spectral behavior of regular Hermitian matrix pencils is examined under certain structure-preserving rank-1 and rank-2 perturbations. Since Hermitian pencils have signs attached to real (and infinite) blocks in canonical form, it is not only the Jordan structure but also this so-called sign characteristic that needs to be examined under perturbation. The observed effects are as follows: Under a rank-1 or rank-2 perturbation, generically the largest one or two, respectively, Jordan blocks at each eigenvalue lambda are destroyed, and if lambda is an eigenvalue of the perturbation, also one new block of size one is created at lambda. If lambda is real (or infinite), additionally all signs at lambda but one or two, respectively, that correspond to the destroyed blocks, are preserved under perturbation. Also, if the potential new block of size one is real, its sign is in most cases prescribed to be the sign that is attached to the eigenvalue lambda in the perturbation.

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.

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.

scholarly journals On the Reduction of Singular Matrix Pencils

1935 ◽  
Vol 4 (2) ◽  
pp. 67-76 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Canonical Form ◽  
Matrix Pencil ◽  
Rational Method ◽  
Singular Matrix ◽  
Matrix Pencils ◽  
Minimal Index

The following rational method of dealing with the reduction of a singular matrix pencil to canonical form has certain advantages. It is based on the principle of vector chains, the length of the chain determining a minimal index. This treatment is analogous to that employed by Dr A. C. Aitken and the author in Canonical Matrices (1932) 45–57, for the nonsingular case. In Theorems 1 and 2 tests are explicitly given for determining the minimal indices. Theorem 2 gives a method of discovering the lowest row (or column) minimal index. Theoretically it should be possible to state a corresponding theorem for each of these indices, not necessarily the lowest, and prior to any reduction of the pencil. This extension still awaits solution.

scholarly journals On the Equivalence of Two Singular Matrix Pencils

1936 ◽  
Vol 4 (4) ◽  
pp. 224-231 ◽  
Author(s):  
J. Williamson
Keyword(s):  
Canonical Form ◽  
Matrix Pencil ◽  
Rational Method ◽  
Singular Matrix ◽  
Matrix Pencils ◽  
Minimal Index

In a recent paper Turnbull, discussing a rational method for the reduction of a singular matrix pencil to canonical form, has shown how the lowest row, or column, minimal index may be determined directly without reducing the pencil to canonical form. It is the purpose of this note to show how all such indices may be determined, and at the same time to give conditions, somewhat simpler than the usual ones, for the equivalence of two matrix pencils.

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