Random vibration of linear systems with singular matrices based on Kronecker canonical forms of matrix pencils

2021 ◽  
Vol 161 ◽  
pp. 107896
Author(s):  
A.D. Karageorgos ◽  
L. Moysis ◽  
V.C. Fragkoulis ◽  
I.A. Kougioumtzoglou ◽  
A.A. Pantelous
Keyword(s):  
Linear Systems ◽  
Random Vibration ◽  
Canonical Forms ◽  
Matrix Pencils ◽  
Singular Matrices

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 Canonical forms for linear systems

1983 ◽  
Vol 50 ◽  
pp. 437-473 ◽  
Author(s):  
D. Prätzel-Wolters
Keyword(s):  
Linear Systems ◽  
Canonical Forms

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.

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