Formulae of partial reduction for linear systems of first order operator equations

Abstract

This paper deals with reduction of non-homogeneous linear systems of first order operator equations with constant coefficients. An equivalent reduced system, consisting of higher order linear operator equations having only one variable and first order linear operator equations in two variables, is obtained by using the rational canonical form.Key words: Linear system of first order operator equations with constant coefficients, n th order linear operator equation with constant coefficients, sum of principal minors, the rational canonical form, the characteristic polynomial

A note on the reduction formulas for some systems of linear operator equations

“…At the beginning of this section we first present two standard lemmas for reduction process using canonical forms. Afterwards we give a brief exposition of two main results from [5] and [6]. where ν = T −1 ϕ is its nonhomogeneous term and z = T −1 x is a column of the unknowns.…”

“…Theorem 3.3. (Theorem 3.4 from [5]) Let us assume that the rational canonical form of the system matrix B has only one block, i.e. that the rational canonical form of B is the companion matrix of the characteristic polynomial ∆ B (λ) = λ n + d 1 λ n−1 + .…”

“…(Theorem 4.1 from[5]) Assume that the system (1) is given in the matrix formA( x) = B x + ϕ,and that matrices B 0 , . .…”

Splitting a linear system of operator equations with constant coefficients: A matrix polynomial approach

“…The strategy which is usually employed is based on the general canonical forms such as Jordan forms [1] and rational canonical forms [11]. A partial reduction procedure based on rational canonical forms has been presented for linear system of operator equations (1) in [11], and recently, the Smith canonical form has been applied for the reduction of the system of integral equations [15,16].…”

“…Consider a linear system of n operator equations with constant coefficients in unknowns x i from [11]:…”