# On Solving Systems of Autonomous Ordinary Differential Equations by Reduction to a Variable of an Algebra

Authors: Alvaro Alvarez-Parrilla; Martín Eduardo Frías-Armenta; Elifalet López-González; Carlos Yee-Romero
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

A new technique for solving a certain class of systems of autonomous ordinary differential equations over𝕂n is introduced (𝕂 being the real or complex field). The technique is based on two observations: (1), if 𝕂n has the structure of certain normed, associative, commutative, and with a unit, algebras 𝔸 over 𝕂, then there is a scheme for reducing the system of differential equations to an autonomous ordinary differential equation on one variable of the algebra; (2) a technique, previously introduced for solving differential equations over ℂ, is shown to work on the class mentioned in the previous paragraph. In particular it is shown that the algebras in question include algebras linearly equivalent to the tensor product of matrix algebras with certain normal forms.


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## Body

## 1. Introduction

Throughout this work,𝕂 will stand for a field, usually the real ℝ or the complex field ℂ.Consider the autonomous ordinary differential equation(1.1)dxdt=f(x),x∈Ω⊂𝕂n,t∈ℝ,
with f:Ω⊂𝕂n→𝕂n having certain regularity conditions.In general the solution is not easy to obtain since this is usually a system of coupled differential equations. There is a vast literature regarding the solution of ordinary differential equations by different means and in particular by techniques utilizinggeneralized analytic functions, see for instance [1–14]. These include applications to the three-dimensional Stokes problem, solutions of planar elliptic vector fields with degeneracies, the Dirichlet problem, multidimensional stationary Schrödinger equation, among others [3, 5, 6, 12, 13]. In particular, the technique that we present is of interest for people working on vector fields with singularities. For instance, in order to gain insight into the behaviour of analytic vector fields, correct visualization of vector fields in the vicinity of their singular set is required, in the case of visualization of two-dimensional complex analytic vector fields with essential singularities the usual methods only provide partial results (see [15–17]), whilst the technique which we promote provides accurate and correct solutions [18, 19]. These questions arise naturally in discrete and continuous dynamical systems (see [20–22]).As a first step in obtaining a solution to (1.1), we notice that, if 𝕂n can be given the structure of a certain algebra 𝔸, it is possible to reduce this system to a single autonomous differential equation
(1.2)dζdt=g~(ζ),ζ∈Ω'⊂𝔸,t∈ℝ,
with g~ being a function 𝔸-differentiable with respect to the variable ζ in the algebra 𝔸.Having done so, we proceed to show that it is possible to solve (1.2) by extending a geometric technique introduced in [18, 19] in the context of complex analytic vector fields related to Newton vector fields which are first studied by [23]. This technique is based upon the construction of two functions, which, are respectively, constant and linear on the trajectories which are the solutions of (1.2).The paper is organized as follows. In Section2 the algebras in question are introduced, in particular we introduce the notion of normed, associative, commutative, and with a unit, finite dimensional algebra 𝔸 over 𝕂, showing in Section 2.1 that these have a first fundamental representation into the algebra of n×n matrices over 𝕂, M(n,𝕂). Furthermore in Section 2.2  normal algebras are defined and their corresponding tensor products are constructed. This is standard material which can be found in [24–26] but is presented here for completeness.In Section3 we give the definition of 𝔸-differentiability, and proceed to show that if the family of matrices {Jf(x):x∈Ω} is linearly equivalent to a subset of an algebra 𝔹 in M(n,𝕂), that is, the image of the first fundamental representation of an algebra 𝔸 with respect to the canonical basis of 𝕂n, that is, 𝔹=R(𝔸), then in fact f is 𝔸-differentiable on Ω. The problem of determining if a map f:Ω⊂𝕂n→𝕂n is 𝔸-differentiable for some algebra 𝔸 is treated indirectly in [27], where the conditions that are given ensure the existence of an algebra 𝔸 such that the set of relations are the generalized Cauchy-Riemann equations for 𝔸, ([27, 28] show that these equations give a criterion for 𝔸-differentiability). Furthermore [29] considers the case when 𝕂=ℂ, where he proves that every analytic map f (in the usual sense) which is 𝔸-differentiable has an expansion in power series (see [29, pp. 646 and 653]).In Section4 we show that for normal algebras it is possible to express the function f(x) in terms of a single variable ζ in the algebra, and hence there is a function g~(ζ) that represents f(x). The analogy being the algebra of complex numbers, where z=x+iy is the variable and the 𝔸-differentiable functions being the analytic functions. We also show that there is a differentiable operator ∂/∂ζ which has the property that ∂g~/∂ζ=g~', where g~' is the Lorch derivative of g~ (see [26]), hence providing a framework for the usual calculus of one variable.In Section5 we start by showing that in this context the differential equation (1.1) takes the form
(1.3)dζdt=g~(ζ),ζ∈Ω′∖𝒮⊂𝔸,t∈ℝ,
with g~ being a function 𝔸-differentiable with respect to the variable ζ in the algebra 𝔸, and 𝒮 being a certain singular set, where the solutions are not defined.We then proceed to show that the geometric technique, introduced in [18, 19], of finding two functions h1 and h2 which are constant and linear on the trajectories ζ(t) which are solutions of the differential equation, can be extended to the case of (1.3). We end the section and the paper with an example.


## 2. Algebras

We introduce𝕂-algebras (see, e.g., [24]).Definition 2.1.
A𝕂-algebra (or algebra over 𝕂) is a finite dimensional 𝕂-linear space 𝔸 on which is defined a bilinear map 𝔸×𝔸→𝔸 that is associative and commutative, and there is a unit element e=e𝔸 in 𝔸 that satisfies ex=xe=x for all x∈A.An elementa∈𝔸 is called regular if there exist a unique element in 𝔸 denoted by a-1∈𝔸 called inverse of a such that a-1a=aa-1=e. An element a∈𝔸 which is not regular is called singular. If a,b∈𝔸 and b is regular, the quotient a/b will mean ab-1.


### 2.1. Algebras and Their Fundamental Representations

We define the first fundamental representation.Ifℬ={β1,…,βn} is an ordered basis of an algebra 𝔸, the product between the elements of ℬ is given by
(2.1)βiβj=∑k=1ncijkβk,
where cijk∈𝕂 for i,j,k∈{1,2,…,n} are called the structure constants of 𝔸. The first fundamental representation of 𝔸 associated to ℬ is the isomorphism R:𝔸→M(n,𝕂) defined by
(2.2)R(x1β1+x2β2+⋯+xnβn)=x1R1+x2R2+⋯+xnRn,
where Ri is the matrix whose entry (j,k) is cijk for i=1,2,…,n. The commutativity and the associativity of 𝔸 are equivalent to the identities  (1)
cijk=cjik, for all i,j,k∈{1,2…,n} and(2)
∑l‍cijlclks=∑l‍cjklcils, for all i,j,k,s∈{1,2…,n}, respectively.UsingR we assign to 𝔸 the norm induced from the operator norm in M(n,𝕂) (see [24]). In this way each algebra is a normed algebra, that is, there exists a norm ∥·∥:𝔸→ℝ satisfying ∥xy∥≤∥x∥∥y∥ for all x,y∈𝔸 and ∥e∥=1.Example 2.2.
Let𝔸 be the linear space ℝ3 with the product between the elements of the standard basis ℬ={e1,e2,e3} given in the following equation:
(2.3)e1e2e3e1e1e2e3e2e2-2e1+2e2+e3-e1+e2+e3e3e3-e1+e2+e3-e1+2e3,
that extends to the product in ℝ3 given by
(2.4)(x,y,z)(u,v,w)=(e1x,e2y,e3z)(e1u,e2v,e3w)=xue1e1+xve1e2+xwe1e3+yue1e2+yve2e2+ywe2e3+zue1e3+zve2e3+xwe3e3=(xu-2yv-yw-zv-zw)e1+(xv-yu-zyv-yw+zv)e2+(zw+zu+yv+yw+zv+zw)e3.The product between the elements ofℬ define the structure constantscijk for i,j,k∈{1,2,3}. So, ℝ3 with the product given is an algebra 𝔸.


### 2.2. Normal Algebras and Their Tensor Products

LetB1,B2,…,Bl be matrices Bi∈M(ki,𝕂), each Bi of one the following four types:
(2.5)(r),(s00⋯01s0001s0⋮⋱000⋯s),(a-bba),(C00⋯0I2C000I2C0⋮⋱000⋯C),
where
(2.6)C=(u-vvu),I2=(1001),r,s,a,b,u,v∈𝕂 and k1+⋯+kl=n. In this case we will say that the matrix B given by
(2.7)B=(B10⋯00B20⋮⋱00⋯Bl),
is in its normal form. We will associate to B an algebra of matrices of dimension n over 𝕂 which contains B and use the following nomenclature: The first block will be called real simple block, the second real Jordan block, the third simple complex block, and the fourth complex Jordan block.Fori=1,2,…,l, let σi:M(ki,𝕂)→M(n,𝕂) be the linear sections defined by substituting the matrix M∈M(ki,𝕂) in the block Bi of the matrix B and taking the other entries as zero. The real Jordan block may be written in the following way Bi=aiDi+Ni, where Di is the identity and Ni is a nilpotent matrix of order ki. The simple complex block may be written in the form Bi=aiDi+biJi, where Di is a diagonal matrix and Ji is a matrix with Ji2=-Di. The complex Jordan blocks may be written in the form Bi=aiDi+biJi+Ni, where Di is the identity and Ji is a matrix with Ji2=-Di and Ni is a nilpotent matrix of order ki/2.We define the matrices{βi,j:1≤i≤l,1≤j≤ki} whose entries are in the block Bi as follows.(i)
IfBi is a real simple block, βi,1:=σi(1), in this case ki=1.(ii)
IfBi is a real Jordan block, βi,1:=σi(Di) and βi,j:=σi(Nij-1) for j=2,…,ki.(iii)
IfBi is a simple complex block, βi,1:=σi(Di) and βi,2:=σi(Ji).(iv)
IfBi is a complex Jordan block, βi,1:=σi(Di), βi,2:=σi(Ji), βi,2j-1:=σi(Ni2j-1), and βi,2j:=βi,2σi(Ni2j-1) for j=2,…,ki/2.Observe that the product of the matricesβi1,j1 and βi2,j2 is the zero matrix if i1≠i2. The products of βi,j1 and βi,j2 for i=1,…,l are as follows.(i)
IfBi is a real simple block, βi,1βi,1=βi,1.(ii)
IfBi is a real Jordan block, then
(2.8)βi,1βi,j1βi,1βi,1βi,j1βi,j2βi,j1βi,j1+j2,
for 2≤j1,j2≤ki, where βi,j1=0 when j1≥ki+1.(iii)
IfBi is a complex simple block, then
(2.9)βi,1βi,2βi,1βi,1βi,2βi,2βi,2-βi,1(iv)
IfBi is a complex Jordan block, then
(2.10)βi,1βi,2βi,2j1-1βi,2j1βi,1βi,1βi,2βi,2j1-1βi,2j1βi,2βi,2-βi,1βi,2j1-βi,2j1-1βi,2j2-1βi,2j2-1βi,2j2βi,2(j1+j2-1)-1βi,2(j1+j2-1)βi,2j2βi,2j2-βi,2j2-1βi,2(j1+j2-1)-βi,2(j1+j2-1)-1,
for 2≤j1,j2≤ki/2, where βi,j=0 when j≥2ki+1.The commutativity of the elements in the set(2.11)ℬ={β1,1,…,β1,k1,…,βl,1,…,βl,kl},
with respect to the matrix product, follows from the well-known result: For a Jordan canonical form D+N the diagonal matrix D commutes with the nilpotent matrix N.Moreover, the𝕂-linear space spanned by ℬ is an 𝕂-algebra, as is claimed in the following proposition.Proposition 2.3.
The setℬ:={β1,1,…,β1,k1,…,βl,1,…,βl,kl} is a base for an n-dimensional linear space which is an algebra 𝔸 with respect to the matrix product, and its first fundamental representation R with respect to ℬ is the identity isomorphism.Proof.
LetR be the first fundamental representation of 𝔸 associated to ℬ. As the algebra 𝔸 is generated by βi,1, βi,2 (considered only when this exists, i.e., ki≥2), and βi,3 (considered only in the case when Bi is a Jordan complex block), in order to prove that R is the identity, we only need to prove that R(βi,j)=βi,j, but for these three cases the equality is trivial. So, R is the identity isomorphism.Definition 2.4.
Given a matrixB∈M(n,𝕂) in its normal form, one will call the algebra in M(n,𝕂), as constructed above, an 𝕂-normal algebra (containingB).Definition 2.5.
Two matrix algebras𝔸1 and 𝔸2 in M(n,𝕂) are linearly equivalent if there exists an invertible matrix B∈M(n,𝕂) such that 𝔸1={BAB-1:A∈𝔸2}.For a proof of the following result, see [30].Proposition 2.6.
Let𝔸 and 𝔹 be algebras. There exists a product in 𝔸⊗𝔹 satisfying
(2.12)(a1⊗b1)(a2⊗b2)=a1a2⊗b1b2,
where a1a2 and b1b2 denote the products in 𝔸 and 𝔹, respectively. The product is associative and e𝔸⊗e𝔹=e𝔸⊗𝔹.Therefore, the finite tensor product of algebras is an algebra.Definition 2.7.
Thecomplexification of an ℝ-algebra 𝔸 is the ℂ-algebra 𝔸⊗ℂ. One calls an algebra which is the complexification of an ℝ-normal algebra a ℂ-normal algebra.As usual, if the context is clear, one will drop theℂ from the name and refer to the ℂ-normal algebra just as a normal algebra. The following proposition and its corollary follow from a straightforward calculation.Proposition 2.8.
Let𝔸 and 𝔹 be p- and q-dimensional matrix algebras in M(n,𝕂) and M(m,𝕂), respectively, and P∈M(n,𝕂) and Q∈M(m,𝕂) be invertible matrices. Then, one has
(2.13)P𝔸P-1⊗Q𝔹Q-1=[P⊗Q][𝔸⊗𝔹][P⊗Q]-1.Corollary 2.9.
The tensor product of matrix algebras linearly equivalent to normal algebras is algebras which are linearly equivalent to the tensor product of normal algebras.So, by Corollary2.9 the algebras linearly equivalent to the tensor product of normal algebras are closed under the tensor product.The following result shows that the first fundamental representation, with respect to an appropriate base, of a tensor product of normal algebras is the inclusion of the algebra in the corresponding matrix space.Proposition 2.10.
Let𝔸 and 𝔹 be p- and q-dimensional 𝕂-algebras, and R1:𝔸→M(p,𝕂) and R2:𝔹→M(q,𝕂) be first fundamental representations associated to the basis 𝒜={α1,…,αp} and ℬ={β1,…,βq}, respectively. Then, R:𝔸⊗𝔹→M(pq,𝕂) defined by R=R1⊗R2 is the first fundamental representation of 𝔸⊗𝔹 associated to the base {αi⊗βj:1≤j≤p,1≤j≤q}.Proof.
Let{α1,…,αp} be a base of 𝔸 and let {β1,…,βq} be a base of 𝔹. We use the notations Ci and Gj for the matrices R1(αi) and R2(βj), respectively, for 1≤i≤p, 1≤j≤q. The set
(2.14){αi⊗βj:1≤i≤p,1≤j≤q}
is a base for 𝔸⊗𝔹 (see [30]). In order to find the structure constants of 𝔸⊗𝔹 we take the products
(2.15)[αi⊗βl][αj⊗βs]=αiαj⊗βlβs=(∑k=1pcijkαk)⊗(∑t=1qdlstβt)=∑k=1p∑t=1qcijkdlst(αk⊗βt),
from which we obtain a first fundamental representation R of A⊗B, where R(αi⊗βl) is the matrix Hil whose entry (kl,js) is given by hil,kt,js:=cijkdlst, where cijk and dlst are the entries (k,j) and (t,s) of Ci and Gl for 1≤i≤p and 1≤l≤q, respectively.
On the other hand we have thatR1(αi)⊗R2(βl)=Ci⊗Gl. The tensor product of Ci and Gl is given by
(2.16)Ci⊗Gl=(ci11(di11di21…diq1⋮di1qdi2q…diqq)⋯cip1(di11di21…diq1⋮di1qdi2q…diqq)⋮ci1p(di11di21…diq1⋮di1qdi2q…diqq)⋯cipp(di11di21…diq1⋮di1qdi2q…diqq)),
from which we see that in the position (kt,js) the element cijkdlst appears. Thus, we have the equalities of matrices Hil=Ci⊗Gl, for 1≤i,l≤n. Therefore, R=R1⊗R2.Corollary 2.11.
If𝔸 is a matrix algebra in M(n,𝕂) which is a tensor product of 𝕂-normal algebras, then there exists a base ℬ of 𝔸 in which the corresponding first fundamental representation R:𝔸→M(n,𝕂) is the identity isomorphism, that is, R(x)=x for all x∈𝔸.Proof.
We have that𝔸=𝔹1⊗⋯⊗𝔹m, where 𝔹i is a 𝕂-normal algebra for every i∈{1,…,m}. Obviously, for every i∈{1,…,m} we can consider a base for 𝔹i as that given in Proposition 2.3, and taking the corresponding tensor products of these basis we obtain a base ℬ for 𝔸. By Propositions 2.3 and 2.10 we have that the first fundamental representation of 𝔸 associated to ℬ is the identity.


## 2.1. Algebras and Their Fundamental Representations

We define the first fundamental representation.Ifℬ={β1,…,βn} is an ordered basis of an algebra 𝔸, the product between the elements of ℬ is given by
(2.1)βiβj=∑k=1ncijkβk,
where cijk∈𝕂 for i,j,k∈{1,2,…,n} are called the structure constants of 𝔸. The first fundamental representation of 𝔸 associated to ℬ is the isomorphism R:𝔸→M(n,𝕂) defined by
(2.2)R(x1β1+x2β2+⋯+xnβn)=x1R1+x2R2+⋯+xnRn,
where Ri is the matrix whose entry (j,k) is cijk for i=1,2,…,n. The commutativity and the associativity of 𝔸 are equivalent to the identities  (1)
cijk=cjik, for all i,j,k∈{1,2…,n} and(2)
∑l‍cijlclks=∑l‍cjklcils, for all i,j,k,s∈{1,2…,n}, respectively.UsingR we assign to 𝔸 the norm induced from the operator norm in M(n,𝕂) (see [24]). In this way each algebra is a normed algebra, that is, there exists a norm ∥·∥:𝔸→ℝ satisfying ∥xy∥≤∥x∥∥y∥ for all x,y∈𝔸 and ∥e∥=1.Example 2.2.
Let𝔸 be the linear space ℝ3 with the product between the elements of the standard basis ℬ={e1,e2,e3} given in the following equation:
(2.3)e1e2e3e1e1e2e3e2e2-2e1+2e2+e3-e1+e2+e3e3e3-e1+e2+e3-e1+2e3,
that extends to the product in ℝ3 given by
(2.4)(x,y,z)(u,v,w)=(e1x,e2y,e3z)(e1u,e2v,e3w)=xue1e1+xve1e2+xwe1e3+yue1e2+yve2e2+ywe2e3+zue1e3+zve2e3+xwe3e3=(xu-2yv-yw-zv-zw)e1+(xv-yu-zyv-yw+zv)e2+(zw+zu+yv+yw+zv+zw)e3.The product between the elements ofℬ define the structure constantscijk for i,j,k∈{1,2,3}. So, ℝ3 with the product given is an algebra 𝔸.


## 2.2. Normal Algebras and Their Tensor Products

LetB1,B2,…,Bl be matrices Bi∈M(ki,𝕂), each Bi of one the following four types:
(2.5)(r),(s00⋯01s0001s0⋮⋱000⋯s),(a-bba),(C00⋯0I2C000I2C0⋮⋱000⋯C),
where
(2.6)C=(u-vvu),I2=(1001),r,s,a,b,u,v∈𝕂 and k1+⋯+kl=n. In this case we will say that the matrix B given by
(2.7)B=(B10⋯00B20⋮⋱00⋯Bl),
is in its normal form. We will associate to B an algebra of matrices of dimension n over 𝕂 which contains B and use the following nomenclature: The first block will be called real simple block, the second real Jordan block, the third simple complex block, and the fourth complex Jordan block.Fori=1,2,…,l, let σi:M(ki,𝕂)→M(n,𝕂) be the linear sections defined by substituting the matrix M∈M(ki,𝕂) in the block Bi of the matrix B and taking the other entries as zero. The real Jordan block may be written in the following way Bi=aiDi+Ni, where Di is the identity and Ni is a nilpotent matrix of order ki. The simple complex block may be written in the form Bi=aiDi+biJi, where Di is a diagonal matrix and Ji is a matrix with Ji2=-Di. The complex Jordan blocks may be written in the form Bi=aiDi+biJi+Ni, where Di is the identity and Ji is a matrix with Ji2=-Di and Ni is a nilpotent matrix of order ki/2.We define the matrices{βi,j:1≤i≤l,1≤j≤ki} whose entries are in the block Bi as follows.(i)
IfBi is a real simple block, βi,1:=σi(1), in this case ki=1.(ii)
IfBi is a real Jordan block, βi,1:=σi(Di) and βi,j:=σi(Nij-1) for j=2,…,ki.(iii)
IfBi is a simple complex block, βi,1:=σi(Di) and βi,2:=σi(Ji).(iv)
IfBi is a complex Jordan block, βi,1:=σi(Di), βi,2:=σi(Ji), βi,2j-1:=σi(Ni2j-1), and βi,2j:=βi,2σi(Ni2j-1) for j=2,…,ki/2.Observe that the product of the matricesβi1,j1 and βi2,j2 is the zero matrix if i1≠i2. The products of βi,j1 and βi,j2 for i=1,…,l are as follows.(i)
IfBi is a real simple block, βi,1βi,1=βi,1.(ii)
IfBi is a real Jordan block, then
(2.8)βi,1βi,j1βi,1βi,1βi,j1βi,j2βi,j1βi,j1+j2,
for 2≤j1,j2≤ki, where βi,j1=0 when j1≥ki+1.(iii)
IfBi is a complex simple block, then
(2.9)βi,1βi,2βi,1βi,1βi,2βi,2βi,2-βi,1(iv)
IfBi is a complex Jordan block, then
(2.10)βi,1βi,2βi,2j1-1βi,2j1βi,1βi,1βi,2βi,2j1-1βi,2j1βi,2βi,2-βi,1βi,2j1-βi,2j1-1βi,2j2-1βi,2j2-1βi,2j2βi,2(j1+j2-1)-1βi,2(j1+j2-1)βi,2j2βi,2j2-βi,2j2-1βi,2(j1+j2-1)-βi,2(j1+j2-1)-1,
for 2≤j1,j2≤ki/2, where βi,j=0 when j≥2ki+1.The commutativity of the elements in the set(2.11)ℬ={β1,1,…,β1,k1,…,βl,1,…,βl,kl},
with respect to the matrix product, follows from the well-known result: For a Jordan canonical form D+N the diagonal matrix D commutes with the nilpotent matrix N.Moreover, the𝕂-linear space spanned by ℬ is an 𝕂-algebra, as is claimed in the following proposition.Proposition 2.3.
The setℬ:={β1,1,…,β1,k1,…,βl,1,…,βl,kl} is a base for an n-dimensional linear space which is an algebra 𝔸 with respect to the matrix product, and its first fundamental representation R with respect to ℬ is the identity isomorphism.Proof.
LetR be the first fundamental representation of 𝔸 associated to ℬ. As the algebra 𝔸 is generated by βi,1, βi,2 (considered only when this exists, i.e., ki≥2), and βi,3 (considered only in the case when Bi is a Jordan complex block), in order to prove that R is the identity, we only need to prove that R(βi,j)=βi,j, but for these three cases the equality is trivial. So, R is the identity isomorphism.Definition 2.4.
Given a matrixB∈M(n,𝕂) in its normal form, one will call the algebra in M(n,𝕂), as constructed above, an 𝕂-normal algebra (containingB).Definition 2.5.
Two matrix algebras𝔸1 and 𝔸2 in M(n,𝕂) are linearly equivalent if there exists an invertible matrix B∈M(n,𝕂) such that 𝔸1={BAB-1:A∈𝔸2}.For a proof of the following result, see [30].Proposition 2.6.
Let𝔸 and 𝔹 be algebras. There exists a product in 𝔸⊗𝔹 satisfying
(2.12)(a1⊗b1)(a2⊗b2)=a1a2⊗b1b2,
where a1a2 and b1b2 denote the products in 𝔸 and 𝔹, respectively. The product is associative and e𝔸⊗e𝔹=e𝔸⊗𝔹.Therefore, the finite tensor product of algebras is an algebra.Definition 2.7.
Thecomplexification of an ℝ-algebra 𝔸 is the ℂ-algebra 𝔸⊗ℂ. One calls an algebra which is the complexification of an ℝ-normal algebra a ℂ-normal algebra.As usual, if the context is clear, one will drop theℂ from the name and refer to the ℂ-normal algebra just as a normal algebra. The following proposition and its corollary follow from a straightforward calculation.Proposition 2.8.
Let𝔸 and 𝔹 be p- and q-dimensional matrix algebras in M(n,𝕂) and M(m,𝕂), respectively, and P∈M(n,𝕂) and Q∈M(m,𝕂) be invertible matrices. Then, one has
(2.13)P𝔸P-1⊗Q𝔹Q-1=[P⊗Q][𝔸⊗𝔹][P⊗Q]-1.Corollary 2.9.
The tensor product of matrix algebras linearly equivalent to normal algebras is algebras which are linearly equivalent to the tensor product of normal algebras.So, by Corollary2.9 the algebras linearly equivalent to the tensor product of normal algebras are closed under the tensor product.The following result shows that the first fundamental representation, with respect to an appropriate base, of a tensor product of normal algebras is the inclusion of the algebra in the corresponding matrix space.Proposition 2.10.
Let𝔸 and 𝔹 be p- and q-dimensional 𝕂-algebras, and R1:𝔸→M(p,𝕂) and R2:𝔹→M(q,𝕂) be first fundamental representations associated to the basis 𝒜={α1,…,αp} and ℬ={β1,…,βq}, respectively. Then, R:𝔸⊗𝔹→M(pq,𝕂) defined by R=R1⊗R2 is the first fundamental representation of 𝔸⊗𝔹 associated to the base {αi⊗βj:1≤j≤p,1≤j≤q}.Proof.
Let{α1,…,αp} be a base of 𝔸 and let {β1,…,βq} be a base of 𝔹. We use the notations Ci and Gj for the matrices R1(αi) and R2(βj), respectively, for 1≤i≤p, 1≤j≤q. The set
(2.14){αi⊗βj:1≤i≤p,1≤j≤q}
is a base for 𝔸⊗𝔹 (see [30]). In order to find the structure constants of 𝔸⊗𝔹 we take the products
(2.15)[αi⊗βl][αj⊗βs]=αiαj⊗βlβs=(∑k=1pcijkαk)⊗(∑t=1qdlstβt)=∑k=1p∑t=1qcijkdlst(αk⊗βt),
from which we obtain a first fundamental representation R of A⊗B, where R(αi⊗βl) is the matrix Hil whose entry (kl,js) is given by hil,kt,js:=cijkdlst, where cijk and dlst are the entries (k,j) and (t,s) of Ci and Gl for 1≤i≤p and 1≤l≤q, respectively.
On the other hand we have thatR1(αi)⊗R2(βl)=Ci⊗Gl. The tensor product of Ci and Gl is given by
(2.16)Ci⊗Gl=(ci11(di11di21…diq1⋮di1qdi2q…diqq)⋯cip1(di11di21…diq1⋮di1qdi2q…diqq)⋮ci1p(di11di21…diq1⋮di1qdi2q…diqq)⋯cipp(di11di21…diq1⋮di1qdi2q…diqq)),
from which we see that in the position (kt,js) the element cijkdlst appears. Thus, we have the equalities of matrices Hil=Ci⊗Gl, for 1≤i,l≤n. Therefore, R=R1⊗R2.Corollary 2.11.
If𝔸 is a matrix algebra in M(n,𝕂) which is a tensor product of 𝕂-normal algebras, then there exists a base ℬ of 𝔸 in which the corresponding first fundamental representation R:𝔸→M(n,𝕂) is the identity isomorphism, that is, R(x)=x for all x∈𝔸.Proof.
We have that𝔸=𝔹1⊗⋯⊗𝔹m, where 𝔹i is a 𝕂-normal algebra for every i∈{1,…,m}. Obviously, for every i∈{1,…,m} we can consider a base for 𝔹i as that given in Proposition 2.3, and taking the corresponding tensor products of these basis we obtain a base ℬ for 𝔸. By Propositions 2.3 and 2.10 we have that the first fundamental representation of 𝔸 associated to ℬ is the identity.


## 3. Differentiability on Algebras

In a paper published in 1893, Sheffers laid a foundation for a theory of analytic functions on algebras, see [27, 29] and references therein.Differentiability on algebras is a stronger concept than the usual differentiability over𝕂n. If f:U⊂𝕂n→𝕂n is a differentiable map in the open set U, we denote by Jf(x) its Jacobian matrix at the point x, in the standard base of 𝕂n. We also use the notation Jℬf(x) for the Jacobian matrix at the point x of f with respect to a base ℬ.The following definition was introduced in [26].Definition 3.1.
Let𝔸 be an algebra and let f:Ω⊂𝔸→𝔸 be a map defined in the open set Ω. One says that f is 𝔸-differentiable at x0∈𝔸 if there exists an element f'(x0)∈𝔸, which one calls the 𝔸-derivative offatx0, satisfying
(3.1)limh→0∥f(x0+h)-f(x0)-f′(x0)h∥∥h∥=0,
where f'(x0)h denotes the product in 𝔸 of f'(x0) with h. If f is 𝔸-differentiable at all the points of Ω, one says that f is 𝔸-differentiable onΩ and one calls the map f' assigning f'(x) to the point x∈Ω the 𝔸-derivative off, or Lorch derivative off.It follows (see, e.g., [26–29]) that a map f:Ω⊂𝔸→𝔸 is 𝔸-differentiable at x0 if and only if Jℬf(x0)∈R(𝔸) and is continuous as a function of x0, where ℬ:={e1,…,en} is a base of 𝔸 and R:𝔸→M(n,𝕂) is the first fundamental representation of the algebra 𝔸 associated to ℬ.In fact, in this case,Jℬf(x) is the image of f'(x) under the map R, that is, if f is 𝔸-differentiable, then
(3.2)Jℬf(x)=∑i=1nui(x)Ri,
where Ri=R(ei) and ui:Ω→𝕂 for i=1,2,…,n.IfΩ':=R(Ω), 𝔹:=R(𝔸), and g:Ω'⊂𝔹→𝔹 is defined by
(3.3)g(y)=R∘f∘R-1(y),
then g is 𝔹-differentiable and its differential at y is given by g'(y)=∑i=1n‍ui(R-1(y))Ri, thus, the relation between the Jacobian matrix of f and the 𝔹-differential of g is Jℬf(x)=g'(R(x)).The matrix equation (3.2) is equivalent to the n2 equations
(3.4)∂fi∂xj=∑l=1nulclji,
for all i,j∈{1,2,…,n}. Using (3.4) and the associativity of the algebra, we can obtain
(3.5)∑i=1nciks∂fi∂xj=∑i=1ncijs∂fi∂xk,
appearing in [29, p. 646].Remark 3.2.
It should be noted that in the context of algebras equations (3.5) play the same role as the Cauchy-Riemann equations in the case of one complex variable and thus serve as a criterion for analyticity, see [27, 28, 31–33].Suppose we consider a basis𝒜={α1,…,αn} of 𝔸, where αi=∑j=1n‍sjiej for i=1,…,n, sji∈𝕂, it can be proved that J𝒜f=S-1(Jf)S, where S=(sij). If we denote by Si the image of αi under the first fundamental representation of 𝔸 associated to 𝒜, we have for i=1,…,n, that Si=S-1(∑j=1n‍sjiRj)S.Remark 3.3.
For the differentiation of algebras the usual properties of differentiation of functions fromℝn to ℝm remain true. Furthermore, the usual rules of differentiation of functions of one variable are satisfied in the case of algebras, therefore polynomial functions, rational functions, and those expressed by means of convergent power series as the exponential, trigonometric, and other usual functions are differentiable in algebras.The following theorem gives conditions that ensure the existence of an algebra𝔸 in which f is 𝔸-differentiable.Theorem 3.4.
Letf:Ω⊂𝕂n→𝕂n be a C1 map defined in the open set Ω. f is 𝔸-differentiable for an algebra 𝔸 if and only if the set of matrices
(3.6){Jf(x):x∈Ω}
is a subset of an algebra 𝔹 which is linearly equivalent to an algebra 𝕋 which is a finite tensor product of normal algebras in M(n,𝕂). Moreover, 𝔸 is a 𝕂-linear space 𝕂n and has a base ℬ such that the image of the first fundamental representation of 𝔸 associated to ℬ is R(𝔸)=𝕋.Proof.
Let𝒜={α1,…,αn} be a base for 𝕋 as given in Corollary 2.11. Then ℬ={Bα1B-1,…,BαnB-1} is a base for 𝔹, where B is a matrix such that 𝔹=B𝕋B-1. Because Jf(x)∈𝔹 for every x∈Ω, we have Jf(x)=∑i=1n‍ui(x)βi, where ui:Ω→𝕂 are functions and βi=BαiB-1.
Now consider the base𝒢={γ1…,γn} of 𝕂n defined by γi=∑j=1n‍bjiej, where B=(bij) and {e1,…,en} is the standard base of 𝕂n. Then, we have that J𝒢f=B-1(Jf)B. Thus,
(3.7)J𝒢f(x)=B-1(∑i=1nui(x)βi)B=B-1(∑i=1nui(x)(Bα1B-1))B=∑i=1nui(x)αi,
in other words J𝒢f(x)∈𝕋, which means that if we define a product between the elements of 𝒢 using the structure constants of the products of the elements of 𝒜, we have that 𝕂n is an algebra 𝔸 such that its first fundamental representation associated to 𝒢 is that given by R(γi)=αi for i=1,…,n. Therefore f is 𝔸-differentiable.


## 4. Reduction to a Variable in the Algebra

Sometimesf=(f1,…,fn):Ω⊂𝕂n→𝕂n can be expressed as a function of a variable ζ=∑i=1n‍xiei of an algebra 𝔸 whose image under the first fundamental representation is the tensor product of normal algebras (as in the previous sections). In this section we show some necessary conditions for this to be true and also allow for the expression of f in terms of ζ.Proposition 4.1.
With the same hypothesis as Theorem3.4, there exists a basis 𝒟={δ1,…,δn} of invertible elements of the algebra 𝔸 such that the following diagram commutes
(4.1)Ω⊂𝕂n→f𝕂nA↓↓AΩ'⊂𝔸→g~𝔸,
where A is the matrix associated to the change of basis ℰ→𝒟, where ℰ={e1,…,en} is the canonical basis of 𝕂n.Proof.
By a change of basis one has the following commutative diagram(4.2)Ω⊂𝕂n→f𝕂nB↓↓BΩ′′⊂𝔸→g𝔸,
where B is the matrix associated to the change of basis from ℰ to ℬ, which was used in the previous theorem, and g=B∘f∘B-1. Now consider a base 𝒟={δ1,…,δn} of regular elements of 𝔸, where without loss of generality δ1=e. Then if A is the matrix associated to the change of basis from ℰ to 𝒟, let g~=A∘f∘A-1. Note that the domain of g~ is Ω'=A(Ω).Theorem 4.2.
Letf:Ω⊂𝕂n→𝕂n be a 𝔸-differentiable map over an algebra 𝔸 with first fundamental representation R(𝔸), then g~=A∘f∘A-1 can be expressed in a variable ζ=∑j=1nyjδj, for some regular basis 𝒟={δ1,…,δn}. Moreover there is a partial differential operator ∂/∂ζ such that g~'(ζ)=(∂g~/∂ζ)(ζ).Proof.
By Proposition4.1 there is a basis 𝒟={δ1,…,δn} of regular elements of the algebra 𝔸, with δ1=e, without loss of generality we may assume that ||δj||=1. Introducing the variable ζ in the base 𝒟 as ζ=∑i=1n‍yiδi=∑i=1n‍xiei, we then have f(x1,…,xn)=g~(ζ), with g~=A∘f∘A-1, where A is the matrix associated to the change of basis ℰ to 𝒟.
Denote byζk¯ the k-conjugate of ζ related to 𝒟, which is defined for k=2,…,n by
(4.3)ζk¯=y1δ1-ykδk+∑j=2j≠knyjδj=-ykδk+∑j=1j≠knyjδj=ζ-2ykδk.
Then we have
(4.4)yke=ζ-ζk¯2δk-1,fork=2,…,n,
and since
(4.5)y1δ1=ζ-∑j=2nyjδj=ζ-∑j=2nζ-ζj¯2,
then,
(4.6)y1e=12((3-n)ζ+∑j=2nζj¯)δ1-1.
Furthermore one can define the differential operators
(4.7)∂∂ζ=1n∑j=1nδj-1∂∂yj,(4.8)∂∂ζk¯=12{δ1-1∂∂y1-δk-1∂∂yk},fork=2,…,n,
which satisfy
(4.9)∂ζ∂ζ=1∂ζk¯∂ζ=n-2nfork=2,…,n,∂ζk¯∂ζk¯=1∂ζ∂ζk¯=∂ζj¯∂ζk¯=0fork=2,…,n,k≠j.
Iff is 𝔸-differentiable, g~ is also 𝔸-differentiable, hence
(4.10)g~′(ζ)=dg~dt(ζ+tδk)|t=0=dg~dt(ζ+tδj)|t=0,∀j,k,
and since
(4.11)dg~dt(ζ+tδk)|t=0=δk-1∂g~∂yk(ζ),
then for k=2,…,n(4.12)∂g~∂ζk¯(ζ)=12∑i=1n{δ1-1∂gi∂y1(ζ)-δk-1∂gi∂yk(ζ)}δi=0,
so the right hand expression of g~(ζ)=∑i=1n‍gi(ζ)δi does not depend on the conjugated variables. In this way we obtain an expression g~(ζ) depending only on ζ, and not on ζk¯ for k=2,…,n.
Moreover, by (4.7), (4.10), and (4.11) one has that
(4.13)g~′(ζ)=∂g~∂y1(ζ)=∂g~∂ζ(ζ).In the following example we show how we can reduce the variables of a map by substituting for variables in an algebra.Example 4.3.
Letf(x1,x2,x3)=(x12,2x1x2,x22+2x1x3). Then
(4.14)Jf=(2x1002x22x102x32x22x1),
the Jf is in normal form, hence f is 𝔸-differentiable. If
(4.15)N=(000100010)
and I is the identity, the multiplication in the associated algebra R(𝔸) is given by multiplication of matrices I, N, and N2 representing, respectively, R(e1),R(e2),andR(e3), where ℰ={e1,e2,e3} is the canonical basis of ℝ3. It is easy to see that e2 and e3 do not have inverse in the algebra 𝔸. Consider the basis of regular elements
(4.16)𝒟={δ1=e1,δ2=e1+e2,δ3=e1+e3},
we see that the matrix associated to the change of basis from ℰ to 𝒟 is
(4.17)A=(1-1-1010001),
so f is transformed to g~=A∘f∘A-1,
(4.18)g~(y1δ1+y2δ2+y3δ3)=(y12-2y22-2y2y3-y32)δ1+2y2(y1+y2+y3)δ2+(y22+2y3(y1+y2+y3))δ3.
Thus
(4.19)ζ=y1δ1+y2δ2+y3δ3
is the variable in the algebra 𝔸. Now consider the conjugates
(4.20)ζ2¯=y1δ1-y2δ2+y3δ3,ζ3¯=y1δ1+y2δ2-y3δ3.
Then
(4.21)y1e=12(ζ2¯+ζ3¯)δ1-1,y2e=12(ζ-ζ2¯)δ2-1,y3e=12(ζ-ζ3¯)δ3-1,
so by substituting this in g~ and simplifying we obtain
(4.22)g~(ζ)=ζ2,
the function in the variable of the algebra.


## 5. Solving Systems of Ordinary Differential Equations by Reduction to One Variable on an Algebra

By following [18, 19], we obtain, as direct corollaries of Theorem 3.4, Proposition 4.1, and Theorem 4.2, the following results.Corollary 5.1.
Let(5.1)f:Ω⊂ℝn→ℝn,g~:Ω′⊂𝔸→𝔸,
be as in Theorem 4.2. Then there exists Φ:Ω'⊂𝔸→𝔸 which is 𝔸-differentiable on Ω′∖𝒮, where 𝒮 is a singular set, such that
(5.2)g~(ζ)=-Φ(ζ)Φ′(ζ),f(x)=-[Jϕ(x)]-1ϕ(x),
with ϕ(x)=(Φ∘A)(x).Proof.
We need to show that there existsΦ:Ω'⊂𝔸→𝔸 such that
(5.3)g~(ζ)=-Φ(ζ)Φ′(ζ).
Noticing that
(5.4)log(Φ(ζ))′=Φ′(ζ)Φ(ζ)=-1g~(ζ),
and by Remark 3.3, it follows that
(5.5)Φ(ζ)=exp[-∫ζdzg~(z)],
for ζ∈Ω′∖𝒮, where
(5.6)𝒮={ζ∈Ω′:g~(ζ)∉𝔸*},
with 𝔸* being the regular elements of 𝔸.Remark 5.2.
Note that𝒮 can be many things. For instance if g~(ζ)=cζ with c∉𝔸*, then 𝒮=𝔸. On the other hand, if g~(ζ) is the polynomial function g(ζ)=cnζn+⋯+c1ζ+c0 with cn singular, then g~(ζ) may be regular for all ζ∈𝔸, (see [26, p. 418]), in which case 𝒮=∅.Remark 5.3.
As a special case one notices that if𝔸=ℂ then 1/g~ will be a complex analytic function on Ω'∖𝒮 hence 𝒮 consists of isolated points (the isolated singularities of 1/g~). This has been studied in [18, 19].Remark 5.4.
In case that𝒮≠Ω′, then Ω'∖𝒮 is an open dense set in Ω'. This is true since the set 𝔸* is an open dense set in 𝔸 hence for a continuous g~ one has that g~-1(𝔸*) is an open dense subset of Ω'.Remark 5.5.
By the previous remark, in case that𝒮≠Ω′, then Φ(ζ) exists for ζ∈Ω'∖𝒮 even though it might be a multivalued function defined on each component of ζ∈Ω'∖𝒮.Remark 5.6.
Further characterization and properties of𝒮 will be studied elsewhere. In what follows we assume that 𝒮≠Ω'.Letx=(x1,x2,…,xn)∈ℝn∖{0}, then consider the projection onto Sn-1(5.7)∏(x)=x∥x∥∈Sn-1.Corollary 5.7.
Letf:Ω⊂ℝn→ℝn be 𝔸-differentiable in an algebra with first fundamental representation R(𝔸). Then the solutions to the differential equation,
(5.8)dxdt=f(x),
correspond to the level curves of the function
(5.9)h1~(x)=Π(ϕ(x)),
moreover the real-valued function,
(5.10)h2~(x)=log∥ϕ(x)∥,
is a linear function of t along the solutions x(t) of (5.8).Proof.
By Theorem3.4, Proposition 4.1, and Theorem 4.2, the solutions x(t) of (5.8) are in correspondence with the solutions ζ(t) of
(5.11)dζdt=g~(ζ).
On the other hand, Corollary 5.1 shows that
(5.12)Φ(ζ(t))=Φ(ζ(t0))exp[-t-t0],
the result follows immediately by applying Π(·) and log∥·∥ to (5.12).Corollary 5.8.
In particular, to visualize the trajectory that passes through the pointx0∈Ω∖A-1(𝒮) at time t0∈ℝ, one needs only to plot the level curve {x∈Ω:h1~(x)=h1~(x0)}. Moreover one can find explicitly the point x(t1), for t1∈ℝ, as the intersection of the level curve h~1(x)=h~1(x0) and the hypersurface h~2(x)=h~2(x0)-t1+t0.Example 5.9.
Consider the function(5.13)f(x,y)=(u(x,y),v(x,y)),
where
(5.14)u(x,y)=16(2+6x+-22+20x+7y4+5x2+8x(-1+y)+y(-4+5y)+5(2+4x+5y)4+5x2+8x(1+y)+y(4+5y)),v(x,y)=16(-4+6y-5(-4+5x+4y)4+5x2+8x(-1+y)+y(-4+5y)+4-7x-20y4+5x2+8x(1+y)+y(4+5y)).
Let
(5.15)A=(1221),
and notice that
(5.16)A·Jf(A-1(x,y))·A-1=(a(x,y)b(x,y)-b(x,y)a(x,y)),
with
(5.17)a(x,y)=1-2x2+(-2+y)2+x(-6+4x+3y)(x2+(-2+y)2)2+x(4x-3(2+y))(x2+(2+y)2)2-2x2+(2+y)2,b(x,y)=4y(x4+(-4+y2)2+2x2(4+y2))2×[48-96x-24x2-16x3-9x4+2x5+2(-3+2x)(4+x2)y2+(3+2x)y4].
Hence A·Jf(A-1(x,y))·A-1 belongs to the normal algebra ℂ, so the previous results are available.
Thus(5.18)g~(x,y)=(A∘f∘A-1)(x,y)=(u1(x,y),v1(x,y)),
with
(5.19)u1(x,y)=-1+x+6-4x-3y2(x2+(-2+y)2)+6-4x+3y2(x2+(2+y)2),v1(x,y)=x(-12+12x+x3)y+2(-2+x2)y3+y5x4+(-4+y2)2+2x2(4+y2)
is just a (complex) analytic function, since
(5.20)Jg~(x,y)=A·Jf(A-1(x,y))·A-1∈R(ℂ),
so by letting z=x+iy, we have
(5.21)g~(z)=u1(z+z¯2,z-z¯2i)+iv1(z+z¯2,z-z¯2i)=2-z2+z34+z2.
By Corollary5.1 one has
(5.22)Φ(z)=exp[-∫zdζg~(ζ)‍],
so that
(5.23)Φ(z)=(z-1-iz-1+i)i(z+1)-1=e2arctan(1-z)z+1.
Note that in ℝ2 one has Π(x,y)=exp[iArg(x,y)] hence the level curves of Π(·) are in correspondence with the level curves of Arg(·). So
(5.24)h(z)=Arg(Φ(z))=-arctan(y1+x)+12log(1-4y2+(-2+x)x+y(2+y))
is a constant of motionh associated to g~(z) and the one associated to f(x,y) is
(5.25)h~(x,y)=(h∘A)(x,y)=-arctan(2x+y1+x+2y)+12log(1-4(2x+y)2+5x2+y(-2+5y)+x(2+8y)).
So by Corollary 5.8 the trajectories associated to
(5.26)(x′(t),y′(t))=f(x(t),y(t)),
are the level curves of h~(x,y). These are presented in Figure 1.Figure 1
Visualization of the trajectories of (5.26). In this case we have plotted level bands of h~(x,y), so that the trajectories are in fact the boundaries between the colored bands. We have also plotted, in white, the trajectory that passes through a singular point corresponding to a pole of the (complex) analytic function g~(z). To exemplify the parametrization, we have plotted in black the points corresponding to t=0,1,2,3 (see Table 1).In order to parametrize the solution we need to calculateh~2(x,y) which turns out to be
(5.27)h~2(x,y)=-arctan(-1+x+2y1-2x-y)+arctan(1-x-2y1+2x+y)-12log((2x+y)2+(1+x+2y)2).
According to Corollary 5.8 we proceeded to calculate the intersection the level curves h~(x,y)=-1.68195 and h~2(x,y)=2.26841-t for t=0,1,2,3,4. The results are shown in Table 1 and as black dots in Figure 1.Table 1
Coordinates of the intersection of the level curvesh~(x,y)=-1.68195 and h~2(x,y)=2.26841-t for t=0,1,2,3,4.
Time (t)
x
t
=
(
x
(
t
)
,
y
(
t
)
)
0
(0.863, −0.783)
1
(0.4808, −0.0710)
2
(0.2002, 0.2548)
3
(0.0678, 0.4654)
4
(91.1109, −52.307)



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*Source: 753916-2012-10-01.xml*