Differential Transcendency in the Theory of Linear Differential Systems with Constant Coefficients

Abstract

We consider a reduction of a nonhomogeneous 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.

1. Introduction

Linear systems with constant coefficients are considered in various fields (see [1–5]). In our paper [5] we use the rational canonical form and a certain sum of principal minors to reduce a linear system of first-order operator equations with constant coefficients to an equivalent, so called partially reduced, system. In this paper we obtain more general results regarding sums of principal minors and a new type of reduction. The obtained formulae of reduction allow some new considerations in connection with Cauchy problem for linear differential systems with constant coefficients and in connection with the differential transcendency of the solution coordinates.

2. Notation

Let us recall some notation. Let 𝐾 be a field, and let 𝐵∈𝐾^𝑛×𝑛 be an 𝑛-square matrix. We denote by(𝛿_𝑘)𝛿_𝑘=𝛿_𝑘(𝐵) the sum of its principal minors of order 𝑘(1≤𝑘≤𝑛), (𝛿^𝑖𝑘)𝛿^𝑖𝑘=𝛿^𝑖𝑘(𝐵) the sum of its principal minors of order 𝑘 containing 𝑖th column (1≤𝑖,𝑘≤𝑛).

Let 𝑣₁,…,𝑣_𝑛 be elements of 𝐾. We write 𝐵^𝑖(𝑣₁,…,𝑣_𝑛)∈𝐾^𝑛×𝑛 for the matrix obtained by substituting ^→𝑣=[𝑣₁…𝑣_𝑛]^𝑇 in place of 𝑖th column of 𝐵. Furthermore, it is convenient to use 𝛿^𝑖𝑘𝐵;^→𝑣=𝛿^𝑖𝑘𝐵;𝑣₁,…,𝑣_𝑛=𝛿^𝑖𝑘𝐵^𝑖𝑣₁,…,𝑣_𝑛,(2.1) and the corresponding vector ^→𝛿_𝑘𝐵;^→𝑣=𝛿^1𝑘𝐵;^→𝑣…𝛿^𝑛𝑘𝐵;^→𝑣^𝑇.(2.2) The characteristic polynomial Δ_𝐵(𝜆) of the matrix 𝐵∈𝐾^𝑛×𝑛 has the following form: Δ_𝐵(𝜆)=det(𝜆𝐼−𝐵)=𝜆^𝑛+𝑑₁𝜆^𝑛−1+⋯+𝑑_𝑛−1𝜆+𝑑_𝑛,(2.3) where 𝑑_𝑘=(−1)^𝑘𝛿_𝑘(𝐵), 1≤𝑘≤𝑛; see [6, page 78].

Denote by 𝐵(𝜆) the adjoint matrix of 𝜆𝐼−𝐵, and let 𝐵₀,𝐵₁,…,𝐵_𝑛−2,𝐵_𝑛−1 be 𝑛-square matrices over 𝐾 determined by 𝐵(𝜆)=adj(𝜆𝐼−𝐵)=𝜆^𝑛−1𝐵₀+𝜆^𝑛−2𝐵₁+⋯+𝜆𝐵_𝑛−2+𝐵_𝑛−1.(2.4)

Recall that(𝜆𝐼−𝐵)𝐵(𝜆)=Δ_𝐵(𝜆)𝐼=𝐵(𝜆)(𝜆𝐼−𝐵).

The recurrence 𝐵₀=𝐼;𝐵_𝑘=𝐵_𝑘−1𝐵+𝑑_𝑘𝐼for1≤𝑘<𝑛 follows from equation 𝐵(𝜆)(𝜆𝐼−𝐵)=Δ_𝐵(𝜆)𝐼; see [6, page 91].

3. Some Results about Sums of Principal Minors

In this section we give two results about sums of principal minors.

Theorem 3.1. For 𝐵∈𝐾^𝑛×𝑛 and ^→𝑣=[𝑣₁…𝑣_𝑛]^𝑇∈𝐾^𝑛×1, one has: 𝛿^𝑖𝑘𝐵;^𝑛_𝑗=1𝑏_1𝑗𝑣_𝑗,…,^𝑛_𝑗=1𝑏_𝑛𝑗𝑣_𝑗+𝛿^𝑖𝑘+1𝐵;𝑣₁,…,𝑣_𝑛=𝛿_𝑘(𝐵)𝑣_𝑖.(3.1)

Remark 3.2. The previous result can be described by 𝛿^𝑖𝑘𝐵;𝐵^→𝑣+𝛿^𝑖𝑘+1𝐵;^→𝑣=𝛿_𝑘(𝐵)𝑣_𝑖,1≤𝑖≤𝑛,(3.2) or simply by the following vector equation: ^→𝛿_𝑘𝐵;𝐵^→𝑣+^→𝛿_𝑘+1𝐵;^→𝑣=𝛿_𝑘(𝐵)^→𝑣.(3.3)

Proof of Theorem 3.1. Let ^→𝑒_𝑠∈𝐾^𝑛×1 denote the column whose only nonzero entry is 1 in 𝑠th position. We also write 𝐵_↓𝑠 for 𝑠th column of the matrix 𝐵 and [𝐵]𝑠̂ for a square matrix of order 𝑛−1 obtained from 𝐵 by deleting its 𝑠th column and row. According to the notation used in (2.1), let [𝐵^𝑖(𝐵_↓𝑠)]𝑠̂ stand for the matrix of order 𝑛−1 obtained from 𝐵 by substituting 𝑠th column 𝐵_↓𝑠 in place of 𝑖th column, and then by deleting 𝑠th column and 𝑠th row of the new matrix. By applying linearity of ^→𝛿_𝑘(𝐵;^→𝑣) with respect to ^→𝑣, we have 𝛿^𝑖𝑘𝐵;𝐵^→𝑣+𝛿^𝑖𝑘+1𝐵;^→𝑣=𝛿^𝑖𝑘𝐵;^𝑛_𝑠=1𝑣_𝑠𝐵_↓𝑠+𝛿^𝑖𝑘+1𝐵;^𝑛_𝑠=1𝑣_𝑠→𝑒_𝑠=_𝑛_𝑠=1𝑣_𝑠𝛿^𝑖𝑘𝐵;𝐵_↓𝑠+^𝑛_𝑠=1𝑣_𝑠𝛿^𝑖𝑘+1𝐵;^→𝑒_𝑠=_𝑛_𝑠=1𝑣_𝑠𝛿^𝑖𝑘𝐵;𝐵_↓𝑠+𝛿^𝑖𝑘+1𝐵;^→𝑒_𝑠=𝑣_𝑖𝛿^𝑖𝑘𝐵;𝐵_↓𝑖+𝛿^𝑖𝑘+1𝐵;^→𝑒_𝑖+^𝑛_{𝑠=1𝑠≠𝑖}𝑣_𝑠𝛿^𝑖𝑘𝐵;𝐵_↓𝑠+𝛿^𝑖𝑘+1𝐵;^→𝑒_𝑠(3.4.) First, we compute 𝑣_𝑖(𝛿^𝑖𝑘(𝐵;𝐵_↓𝑖)+𝛿^𝑖𝑘+1(𝐵;^→𝑒_𝑖))=𝑣_𝑖(𝛿^𝑖𝑘(𝐵)+𝛿_𝑘([𝐵]𝑖)̂)=𝑣_𝑖𝛿_𝑘(𝐵). Then, it remains to show that ∑^{𝑛𝑠=1,𝑠≠𝑖}𝑣_𝑠(𝛿^𝑖𝑘(𝐵;𝐵_↓𝑠)+𝛿^𝑖𝑘+1(𝐵;^→𝑒_𝑠))=0.
It suffices to prove that each term in the sum is zero, that is, 𝛿^𝑖𝑘𝐵;𝐵_↓𝑠+𝛿^𝑖𝑘+1𝐵;^→𝑒_𝑠=0,for𝑠≠𝑖.(3.5) Suppose that 𝑠≠𝑖. We now consider minors in the sum 𝛿^𝑖𝑘(𝐵;𝐵_↓𝑠). All of them containing 𝑠th column are equal to zero, so we deduce 𝛿^𝑖𝑘𝐵;𝐵_↓𝑠=𝛿^𝑖𝑘𝐵^𝑖𝐵_↓𝑠=𝛿^𝑖𝑘𝐵^𝑖𝐵_↓𝑠𝑠̂.(3.6) If 𝑠≠𝑖, then each minor in the sum 𝛿^𝑖𝑘+1(𝐵;^→𝑒_𝑠) necessarily contains 𝑠th row and 𝑖th column. By interchanging 𝑖th and 𝑠th column, we multiply each minor by −1. We now proceed by expanding these minors along 𝑠th column to get −1 times the corresponding 𝑘th order principal minors of matrix 𝐵^𝑖(𝐵_↓𝑠) which do not include 𝑠th column. Hence, 𝛿^𝑖𝑘+1(𝐵;^→𝑒_𝑠)=−𝛿^𝑖𝑘([𝐵^𝑖(𝐵_↓𝑠)]𝑠)̂, and the proof is complete.

In the following theorem, we give some interesting correspondence between the coefficients 𝐵_𝑘 of the matrix polynomial 𝐵(𝜆)=adj(𝜆𝐼−𝐵) and sums of principal minors ^→𝛿_𝑘+1(𝐵;^→𝑣), 0≤𝑘<𝑛.

Theorem 3.3. Given an arbitrary column [𝑣₁…𝑣_𝑛]^𝑇∈𝐾^𝑛×1, it holds 𝐵_𝑘→𝑣=𝐵_𝑘⎡⎢⎢⎢⎢⎢⎢⎣𝑣₁𝑣₂𝑣⋮_𝑛⎤⎥⎥⎥⎥⎥⎥⎦=(−1)_𝑘⎡⎢⎢⎢⎢⎢⎢⎢⎣𝛿𝛿^1𝑘+1_2𝑘+1𝐵;𝑣₁,…,𝑣_𝑛𝛿_𝑛𝑘+1𝐵;⋮𝑣₁,…,𝑣_𝑛𝐵;𝑣₁,…,𝑣_𝑛⎤⎥⎥⎥⎥⎥⎥⎥⎦=(−1)^𝑘→𝛿_𝑘+1𝐵;^→𝑣.(3.7)

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−1𝐵+𝑑_𝑘𝐼, by the vector ^→𝑣, we obtain that 𝐵_𝑘→𝑣=𝐵_𝑘−1𝐵^→𝑣+𝑑_𝑘→𝑣=_(IH)(−1)^→𝑘−1𝛿_𝑘𝐵;𝐵^→=(−1)_𝑘−1𝑣+𝑑_𝑘→𝑣^→𝛿_𝑘𝐵;𝐵^→𝑣−𝛿_𝑘→𝑣=₍₅₎(−1)^𝑘→𝛿_𝑘+1𝐵;^→𝑣.(3.8)

Remark 3.4. Theorem 3.3 seems to have an independent application. Taking ^→𝑣=^→𝑒_𝑗,1≤𝑗≤𝑛, we prove formulae (8)–(10) given in [7].

4. Formulae of Total Reduction

We can now obtain a new type of reduction for the linear systems with constant coefficients from [5] applying results of previous section. For the sake of completeness, we introduce some definition.

Let 𝐾 be a field, 𝑉 a vector space over field 𝐾, and let 𝐴∶𝑉→𝑉 be a linear operator. We consider a linear system of first-order 𝐴-operator equations with constant coefficients in unknowns 𝑥_𝑖, 1≤𝑖≤𝑛, is 𝐴𝑥₁=𝑏₁₁𝑥₁+𝑏₁₂𝑥₂+⋯+𝑏_1𝑛𝑥_𝑛+𝜑₁𝐴𝑥₂=𝑏₂₁𝑥₁+𝑏₂₂𝑥₂+⋯+𝑏_2𝑛𝑥_𝑛+𝜑₂𝐴𝑥_𝑛⋮=𝑏_𝑛1𝑥₁+𝑏_𝑛2𝑥₂+⋯+𝑏_𝑛𝑛𝑥_𝑛+𝜑_𝑛(4.1) for 𝑏_𝑖𝑗∈𝐾 and 𝜑_𝑖∈𝑉. We say that 𝐵=[𝑏_𝑖𝑗]^{𝑛𝑖,𝑗=1}∈𝐾^𝑛×𝑛 is the system matrix, and 𝜑^→=[𝜑₁…𝜑_𝑛]^𝑇∈𝑉^𝑛×1 is the free column.

Let ^→𝑥=[𝑥₁…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(𝑥₁)…𝐴(𝑥_𝑛)]^𝑇. Then system (4.1) can be written in the following vector form: 𝐴^→^→𝑥=𝐵^→𝑥+𝜑^→.(4.2) Any column ^→𝑣∈𝑉^𝑛×1 which satisfies the previous system is its solution.

Powers of operator 𝐴 are defined as usual 𝐴^𝑖=𝐴^𝑖−1∘𝐴 assuming that 𝐴⁰=𝐼∶𝑉→𝑉 is the identity operator. By 𝑛th order linear 𝐴-operator equation with constant coefficients, in unknown 𝑥, we mean 𝐴^𝑛(𝑥)+𝑏₁𝐴^𝑛−1(𝑥)+⋯+𝑏_𝑛𝐼(𝑥)=𝜑,(4.3) where 𝑏₁,…,𝑏_𝑛∈𝐾 are coefficients and 𝜑∈𝑉. Any vector 𝑣∈𝑉 which satisfies (4.3) is its solution.

The following theorem separates variables of the initial system.

Theorem 4.1. Assume that the linear system of first-order 𝐴-operator equations is given in the form (4.2), 𝐴^→(^→𝑥)=𝐵^→𝑥+𝜑^→, and that matrices 𝐵₀,…,𝐵_𝑛−1 are coefficients of the matrix polynomial 𝐵(𝜆)=adj(𝜆𝐼−𝐵). Then it holds the following: Δ^𝐵^→𝐴^→𝑥=^𝑛_𝑘=1𝐵_→𝑘−1𝐴^𝑛−𝑘𝜑^→.(4.4)

Proof. Let 𝐿_𝐵∶𝑉^𝑛×1→𝑉^𝑛×1 be a linear operator defined by 𝐿_𝐵(^→𝑥)=𝐵^→𝑥. Replacing 𝜆𝐼 by 𝐴^→ in the equation Δ_𝐵(𝜆)𝐼=𝐵(𝜆)(𝜆𝐼−𝐵), we obtain that Δ_𝐵^→𝐴=𝐵𝐴^→∘𝐴^→−𝐿_𝐵Δ,hence_𝐵^→𝐴^→𝑥=𝐵𝐴^→𝐴^→−𝐿_𝐵^→=𝑥𝐵𝐴^→𝐴^→^→𝑥−𝐵^→𝑥=₍₈₎𝐵𝐴^→𝜑^→=^𝑛_𝑘=1𝐵_→𝑘−1𝐴^𝑛−𝑘𝜑^→.(4.5)

The next theorem is an operator generalization of Cramer's rule.

Theorem 4.2 (the theorem of total reduction-vector form). Linear system of first-order 𝐴-operator equations (4.2) can be reduced to the system of 𝑛th order 𝐴-operator equations Δ^𝐵^→𝐴^→𝑥=^𝑛_𝑘=1(−1)^→𝑘−1𝛿_𝑘𝐵;^→𝐴^𝑛−1𝜑^→.(4.6)

Proof. It is an immediate consequence of Theorems 4.1 and 3.3 as follows: Δ_𝐵^→𝐴^→𝑥=_𝑛(10)_𝑘=1𝐵_→𝑘−1𝐴^𝑛−𝑘𝜑^→=_𝑛(6)_𝑘=1(−1)^→𝑘−1𝛿_𝑘𝐵;^→𝐴^𝑛−𝑘𝜑^→.(4.7)

We can now rephrase the previous theorem as follows.

Theorem 4.3 (the theorem of total reduction). Linear system of first-order 𝐴-operator equations (4.1) implies the system, which consists of 𝑛th order 𝐴-operator equations as follows: ΔΔ_𝐵𝐵(𝐴)𝑥₁=^𝑛_𝑘=1(−1⋮)^𝑘−1𝛿^1𝑘𝐵;𝐴^{→𝑛−𝑘}𝜑^→(𝐴)𝑥_𝑖=^𝑛_𝑘=1(−1)^𝑘−1𝛿^𝑖𝑘𝐵;^→Δ_𝐵⋮𝐴^𝑛−𝑘𝜑^→(𝐴)𝑥_𝑛=^𝑛_𝑘=1(−1)^𝑘−1𝛿^𝑛𝑘𝐵;^→𝐴^𝑛−𝑘𝜑^→.(4.8)

Remark 4.4. System (4.8) has separated variables, and it is called totally reduced. The obtained system is suitable for applications since it does not require a change of base. This system consists of 𝑛th order linear 𝐴-operator equations which differ only in the variables and in the nonhomogeneous terms.
Transformations of the linear systems of operator equations into independent equations are important in applied mathematics [1]. In the following two sections: we apply our theorem of total reduction to the specific linear operators 𝐴.

5. Cauchy Problem

Let us assume that 𝐴 is a differential operator on the vector space of real functions and that system (4.1) has initial conditions 𝑥_𝑖(𝑡₀)=𝑐_𝑖, for 1≤𝑖≤𝑛. Then the Cauchy problem for system (4.1) has a unique solution. Using form (4.2), we obtain additional 𝑛−1 initial conditions of 𝑖th equation in system (4.8). Consider 𝐴^𝑗𝑥_𝑖𝑡₀=𝐵^𝑗→𝑥𝑡₀_𝑖+^𝑗−1_𝑘=0𝐵^{→𝑗−1−𝑘}𝐴^𝑘𝜑^→𝑡₀_𝑖,1≤𝑗≤𝑛−1,(5.1) where []_𝑖 denotes 𝑖th coordinate. Then each equation in system (4.8) has a unique solution under given conditions and additional conditions (5.1), and these solutions form a unique solution to system (4.1). Therefore, formulae (4.8) can be used for solving systems of differential equations.

It is worth pointing out that the above method can be also extended to systems of difference equations.

6. Differential Transcendency

Now suppose that 𝑉 is the vector space of meromorphic functions over the complex field 𝐶 and that 𝐴 is a differential operator, 𝐴(𝑥)=(𝑑/𝑑𝑧)(𝑥). Let us consider system (4.1) under these assumptions.

Recall that a function 𝑥∈𝑉 is differentially algebraic if it satisfies a differential algebraic equation with coefficients from 𝐶; otherwise, it is differentially transcendental (see [2–4, 8–10]).

Let us consider nonhomogenous linear differential equation of 𝑛th order in the form (4.3), where 𝑏₁,…,𝑏_𝑛∈𝐶 are constants and 𝜑∈𝑉. If 𝑥 is differentially transcendental then Δ_𝐵(𝐴)(𝑥) is also a differentially transcendental function. On the other hand, if 𝜑 is differentially transcendental, then, based on Theorem 2.8. from [10], the solution 𝑥 of (4.3) is a differentially transcendental function. Therefore, we obtain the equivalence.

Theorem 6.1. Let 𝑥 be a solution of (4.3), and then 𝑥 is a differentially transcendental function if and only if 𝜑 is a differentially transcendental function.

We also have the following statement about differential transcendency.

Theorem 6.2. Let 𝜑_𝑗 be the only differentially transcendental component of the free column 𝜑^→. Then for any solution ^→𝑥 of system (4.2), the corresponding entry 𝑥_𝑗 is also a differentially transcendental function.

Proof. The sum ∑^𝑛𝑘=1(−1)^𝑘−1𝛿^𝑗𝑘(𝐵;𝐴^{→𝑛−𝑘}(𝜑^→)) must be a differentially transcendental function. The previous theorem applied to the following equation∶Δ_𝐵(𝐴)𝑥_𝑗=^𝑛_𝑘=1(−1)^𝑘−1𝛿^𝑗𝑘𝐵;^→𝐴^𝑛−𝑘𝜑^→(6.1) implies that 𝑥_𝑗 is a differentially transcendental function too.

Let us consider system (4.1), and let 𝜑₁ be the only differentially transcendental component of the free column 𝜑^→. Then, the coordinate 𝑥₁ is a differentially transcendental function too. Whether the other coordinates 𝑥_𝑘 are differentially algebraic depends on the system matrix 𝐵. From the formulae of total reduction and Theorem 6.1. we obtain the following statement.

Theorem 6.3. Let 𝜑₁ be the only differentially transcendental component of the free column 𝜑^→ of system (4.1). Then the coordinate 𝑥_𝑘, 𝑘≠1, of the solution ^→𝑥, is differentially algebraic if and only if in the sum ∑^𝑛𝑗=1(−1)^𝑗−1𝛿^𝑘𝑗(𝐵;𝐴^{→𝑛−𝑗}(𝜑^→)) appears no function 𝐴^𝑛−𝑗(𝜑₁)(𝑗=1,…,𝑛).

Example 6.4. Let us consider system (4.1) in the form (4.8) in dimensions 𝑛=2 and 𝑛=3 with 𝜑₁ as the only differentially transcendental component. The function 𝑥₁ is differentially transcendental. For 𝑛=2 the function 𝑥₂ is differentially algebraic if and only if 𝑏₂₁=0. For 𝑛=3 the function 𝑥₂ is differentially algebraic if and only if 𝑏₂₁=0∧𝑏₃₁⋅𝑏₂₃=0 and the function 𝑥₃ is differentially algebraic iff 𝑏₃₁=0∧𝑏₃₁⋅𝑏₂₂=0.

Let us emphasize that if we consider two or more differentially transcendental components of the free column 𝜑^→, then the differential transcendency of the solution coordinates also depends on some kind of their differential independence (see e.g., [8]).

Acknowledgment

Research is partially supported by the Ministry of Science and Education of the Republic of Serbia, Grant no. 174032.