On the Kronecker canonical form of a full row rank matrix pencil and applications

The Kronecker canonical form (KCF) of matrix pencils plays an important role in many fields such as systems control and differential–algebraic equations. In this article, we compute a finite and infinite Jordan chain and also a singular chain of vectors corresponding to a full row rank matrix pencil using an extended algorithm, first introduced by Jones (1999, Ph.D. Thesis, Department of Mathematics, Loughborough University of Technology, Loughborough, UK). The proposed method exploits these vectors forming the chains corresponding to the finite and infinite eigenvalues and to the right minimal indices of the pencil. This leads to the computation of two transformation matrices for obtaining under strict equivalence the KCF of the pencil. An application to the study of homogeneous linear rectangular descriptor systems is considered and closed form solutions are obtained in terms of these two transformation matrices. All the results are illustrated with an example.

A block form of a singular pencil of operators and a method of obtaining it

A block form of a singular operator pencil $\lambda A+B$, where $\lambda$ is a complex parameter, and the linear operators $A$, $B$ act in finite-dimensional spaces, is described. An operator pencil $\lambda A+B$ is called regular if $n = m = rk(\lambda A+B)$, where $rk(\lambda A+B)$ is the rank of the pencil and $m$, $n$ are the dimensions of spaces (the operators map an $n$-dimensional space into an $m$-dimensional one); otherwise, if $n \ne m$ or $n = m$ and $rk(\lambda A+B)<n$, the pencil is called singular (irregular). The block form (structure) consists of a singular block, which is a purely singular pencil, i.e., it is impossible to separate out a regular block in this pencil, and a regular block. In these blocks, zero blocks and blocks, which are invertible operators, are separated out. A method of obtaining the block form of a singular operator pencil is described in detail for two special cases, when $rk(\lambda A+B) = m < n$ and $rk(\lambda A+B) = n < m$, and for the general case, when $rk(\lambda A+B) < n, m$. Methods for the construction of projectors onto subspaces from the direct decompositions, relative to which the pencil has the required block form, are given. Using these projectors, we can find the form of the blocks and, accordingly, the block form of the pencil. Examples of finding the block form for the various types of singular pencils are presented. To obtain the block form, in particular, the results regarding the reduction of a singular pencil of matrices to the canonical quasidiagonal form, which is called the Weierstrass-Kronecker canonical form, are used. Also, methods of linear algebra are used. The obtained block form of the pencil and the corresponding projectors can be used to solve various problems. In particular, it can be used to reduce a singular semilinear differential-operator equation to the equivalent system of purely differential and purely algebraic equations. This greatly simplifies the analysis and solution of differential-operator equations.

On the Existence and Uniqueness of the Solution of Linear Fractional Differential-Algebraic System

The existence and uniqueness of the solution of a new kind of system—linear fractional differential-algebraic equations (LFDAE)—are investigated. Fractional derivatives involved are under the Caputo definition. By using the tool of matrix pair, the LFDAE in which coefficients matrices are both square matrices have unique solution under the condition that coefficients matrices make up a regular matrix pair. With the help of equivalent transformation and Kronecker canonical form of the coefficients matrices, the sufficient condition for existence and uniqueness of the solution of the LFDAE in which coefficients matrices are both not square matrices is proposed later. Two examples are given to justify the obtained theorems in the end.

Smith form and Kronecker canonical form

This chapter treats matrix polynomials with quaternion coefficients. A diagonal form (known as the Smith form), which asserts that every quaternion matrix polynomial can be brought to a diagonal form under pre- and postmultiplication by unimodular matrix polynomials, is proved for such polynomials. In contrast to matrix polynomials with real or complex coefficients, a Smith form is generally not unique. For matrix polynomials of first degree, a Kronecker form—the canonical form under strict equivalence—is available, which this chapter presents with a complete proof. Furthermore, the chapter gives a comparison for the Kronecker forms of complex or real matrix polynomials with the Kronecker forms of such matrix polynomials under strict equivalence using quaternion matrices.