On the Kronecker Canonical Form of Singular Mixed Matrix Pencils

2017 ◽  
Vol 55 (3) ◽  
pp. 2134-2150 ◽  
Author(s):  
Satoru Iwata ◽  
Mizuyo Takamatsu
Keyword(s):  
Canonical Form ◽  
Mixed Matrix ◽  
Matrix Pencils ◽  
Kronecker Canonical Form

scholarly journals Regularizing algorithm for mixed matrix pencils

2017 ◽  
Vol 2 (1) ◽  
pp. 123-130 ◽  
Author(s):  
Tetiana Klymchuk
Keyword(s):  
Canonical Form ◽  
Numerical Stability ◽  
Matrix Pencil ◽  
Mixed Matrix ◽  
Complex Matrices ◽  
Regularizing Algorithm ◽  
Unitary Transformations ◽  
Nonsingular Matrices ◽  
Matrix Pencils ◽  
Square Complex

AbstractP. Van Dooren (1979) constructed an algorithm for computing all singular summands of Kronecker’s canonical form of a matrix pencil. His algorithm uses only unitary transformations, which improves its numerical stability. We extend Van Dooren’s algorithm to square complex matrices with respect to consimilarity transformations $\begin{array}{} \displaystyle A \mapsto SA{\bar S^{ - 1}} \end{array}$ and to pairs of m × n complex matrices with respect to transformations $\begin{array}{} \displaystyle (A,B) \mapsto (SAR,SB\bar R) \end{array}$, in which S and R are nonsingular matrices.

Mixed matrix pencils: Nonstandard involutions

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Canonical Form ◽  
The Other ◽  
Canonical Forms ◽  
Quaternion Matrix ◽  
Mixed Matrix ◽  
Quaternion Matrices ◽  
Matrix Pencils

This chapter also studies the canonical forms of mixed quaternion matrix pencils, i.e., such that one of the two matrices is φ‎-hermitian and the other is φ‎-skewhermitian, with respect to simultaneous φ‎-congruence. It starts by formulating the canonical form for φ‎-hsk matrix pencils under strict equivalence. Other canonical forms of mixed matrix pencils are developed with respect to strict equivalence. As an application, this chapter provides canonical forms of quaternion matrices under φ‎-congruence. As in the preceding chapter, this chapter also fixes a nonstandard involution φ‎ throughout and a quaternion β‎(φ‎) such that φ‎=(β‎(φ‎)) = −β‎(φ‎) and ∣β‎(φ‎)∣ = 1.

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.

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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