Differential transcendency in the theory of linear differential systems with constant coefficients

In this paper we consider a reduction of a non-homogeneous linear system of first order operator equations to a totally reduced system. Obtained results are applied to Cauchy problem for linear differential systems with constant coefficients and to the question of differential transcendency.


Introduction
Let K be a field, V a vector space over K and let A : V −→ V be a linear operator. Linear system of first order A-operator equations with constant coefficients in unknowns x i , 1 ≤ i ≤ n, is: for b ij ∈ K and ϕ i ∈ V . We say that B = [b ij ] n i,j=1 ∈ K n×n is the system matrix, and ϕ = [ϕ 1 . . . ϕ n ] T ∈ V n×1 is the free column. If ϕ = 0 system (1) is called homogeneous. Otherwise it is called non-homogeneous. Let x = [x 1 . . . x n ] T be a column of unknowns and A : V n×1 −→ V n×1 be a vector operator defined componentwise A( x) = [A(x 1 ) . . . A(x n )] T . Then system (1) can be written in the following vector form: A solution of vector equation (2) is any column v ∈ V n×1 which satisfies (2). Powers of operator A are defined as usual A i = A i−1 • A assuming that A 0 = I : V −→ V is the identity operator. By n th order linear A-operator equation with constant coefficients, in unknown x, we mean: where b 1 ,. . ., b n ∈ K are coefficients and ϕ is non-homogeneous term. If ϕ = 0 equation (3) is called homogeneous. Otherwise it is non-homogeneous. A solution of equation (3) is any vector v ∈ V which satisfies (3).

Notation
Let M ∈ K n×n be an n-square matrix. We denote by: The characteristic polynomial ∆ B (λ) of the matrix B ∈ K n×n has the form Denote byB(λ) the adjoint matrix of λI − B and let B 0 , B 1 , . . . , B n−2 , B n−1 be n-square matrices over

Main Results
The primary purpose of our paper is to reduce initial system (2) to a system of higher order linear A-operator equations, called totally reduced system. It has separated variables and symmetrical form.
The following theorem will be useful in proving our main result.
Theorem 3.1 Assume that the linear system of first order A-operator equations is given in the form (2), A( x) = B x+ ϕ, and that matrices B 0 , . . . , B n−1 are coefficients of the matrix polynomialB(λ) = adj(λI−B). Then it holds: In addition to the notation given in relation (4), we also use δ Lemma 3.2 Using the above mentioned notation and assuming [v 1 . . . v n ] T ∈ K n×1 , we have: Remark 3.3 The previous result can be described by: or simply by the following vector equation: (8) Proof of Lemma 3.2. Let e s ∈ K n×1 denote the column whose only nonzero entry is 1 in s th position. We also write B ↓s for s th column of the matrix B and [B]ŝ for a square matrix of order n−1 obtained from B by deleting its s th column and row. According to the notation used in (4), let [B i (B ↓s )]ŝ stand for the matrix of order n−1 obtained from B by substituting s th column B ↓s in place of i th column, and then by deleting s th column and s th row of the new matrix. By applying linearity of δ k B; v with respect to v, we have: Then, it remains to show that Proof. The proof proceeds by induction on k. It is being obvious for k = 0. Assume, as induction hypothesis (IH), that the statement is true for k−1. Multiplying the right side of the equation B k = B k−1 B + d k I, by the vector v, we obtain: Remark 3.5 The preceding Lemma 3.4 seems to have an independent application. Taking v = e j , 1 ≤ j ≤ n, we prove formulae (8) − (10) given by [1].
Theorem 3.6 (The main theorem for totally reduced systems -vector form) Linear system of first order A-operator equations (2) can be reduced to the system of n th order A-operator equations: Proof. It is an immediate consequence of Theorem 3.1 and Lemma 3.4: We can now rephrase the previous Theorem as follows.
Theorem 3.7 (The main theorem for totally reduced systems) Linear system of first order A-operator equations (1) implies the system, which consists of n th order A-operator equations: (11) Corollary 3.8 If we take A = 0, Theorem 3.7 is equivalent to Cramer's rule.
Remark 3.9 System (11) has separated variables and it is called totally reduced. This system consists of n th order linear A-operator equations which differ only in variables and in the non-homogeneous terms.
Remark 3.10 Formulae of total and partial reduction [7] could be of interest in the theory of linear control systems [2] and also in problems concerning systems invariants under the action of the general linear group GL(n, K), see [4] and [5].
Applications 1 0 Assume that A is the differential operator on the vector space of real functions. Let us consider system (1) with initial conditions x i (t 0 ) = c i , 1 ≤ i ≤ n. The Cauchy problem for system (1) has a unique solution. Using form (2), we obtain additional n−1 initial conditions of i th equation in system (11): where [ ] i denotes i th coordinate. Then each equation in system (11) has a unique solution under given conditions and additional conditions (12), and these solutions form a unique solution to system (1). Therefore, formulae (11) can be used for solving systems of differential equations. Let us emphasize that we can extend this consideration to systems of difference equations and whenever the above method can be generalized.
2 0 Now suppose that V is the vector space of meromorphic functions over the complex field and that A is a differential operator, i.e. A(w) = d d z (w). Let us consider system (1) under these assumptions. Recall that a function w ∈ V is differentially transcendental if it does not satisfy any algebraic differential equation. We prove the following theorem.
Theorem 3.11 Let ϕ j be the only differentially transcendental component of the free column ϕ. Then for any solution x of system (1), corresponding entry x j is also differentially transcendental function. Proof. We can see that the sum n k=1 (−1) k−1 δ j k B; A n−k ( ϕ) is a differentially transcendental function. According to Theorem 2.8 proved by [6], the following equation: yields that x j is a differentially transcendental function too. 2

Examples
Finally, we can illustrate our results with two special cases of n = 2 and n = 3. (13) Applying Theorem 3.7 the totally reduced system of system (13) has the form: where we take into consideration only framed minors.
Example 4.2 Consider the linear system of the first order A-operator equations, in unknowns x 1 , x 2 , x 3 : Applying Theorem 3.7 the totally reduced system of system (14) has the form: where we calculate only sums of the framed minors.