In this paper we propose a definition of determinant for quaternionic
polynomial matrices. This definition is later used in the study
of stability of linear quaternionic systems within the behavioral setting

Pereira, Ricardo

Rocha, Paula

English

Repositório Institucional da Universidade de Aveiro

Stability of quaternionic systems: adeterminantal approachRicardo Pereira and Paula RochaDepartment of MathematicsUniversity of Aveiro{ricardo, procha}@mat.ua.ptAbstractIn this paper we propose a definition of determinant for quater-nionic polynomial matrices. This definition is later used in the studyof stability of linear quaternionic systems within the behavioral set-ting.1 IntroductionThe quaternions, introduced by Hamilton in 1843, may be favorably used todescribe phenomena occurring in areas such as electromagnetism and quan-tum physics [11], by means of a compact notation that leads to a higherefficiency in computational terms [5]. In particular, they are a powerful toolin the description of rotations. It is not uncommon to find situations, es-pecially in robotics, where the rotation of a rigid body depends on time,and this dynamics is advantageously written in terms of quaternionic dif-ferential or difference equations. The attempt to analyse and to control thecorresponding dynamics motivates the study of such equations from a systemtheoretic point of view.As is well known, the asymptotical stability (henceforth simply referred asstability) of a discrete-time state space system x(t + 1) = Ax(t) with realor complex coefficients, is characterized by the location of the eigenvalues ofthe system matrix A, i.e., of the zeros of the determinant of the polynomialmatrix sI − A.1Theorem 1. [6] The state system x(t + 1) = Ax(t) is stable if and only ifdet(λI − A) 6= 0, ∀λ ∈ C such that |λ| ≥ 1.In a more general setting, the growth of the solutions (or trajectories) of alinear higher order difference equation with constant real or complex (square)matrix coefficientsRpw(t+ p) + · · ·+R1w(t+ 1) +R0w(t) = 0, (1)can also be characterized in terms of the zeros of the determinant of thepolynomial matrix R(s) := Rpsp + · · ·+R1s+R0.Theorem 2. [8] The system described by (1) is stable if and only if detR(λ) 6=0, ∀λ ∈ C such that |λ| ≥ 1.When trying to generalize these results to systems with quaternionic co-efficients, one is confronted with the lack of a notion of determinant forquaternionic polynomial matrices. Indeed, due to the non-commutativity ofthe field of quaternions, the determinant of quaternionic matrices cannot bedefined as in the commutative (eg, real or complex) case. Several definitionshave been proposed for matrices over the quaternionic skew field [1], but thiswork has not been extended to the polynomial ring case.In this paper we try to fill in this gap by proposing a concept of determinantfor quaternionic polynomial matrices. This concept is later used in the studyof stability of linear quaternionic systems.2 PreliminariesWe first introduce some preliminary concepts on quaternions and quater-nionic polynomials.The setH = {a+ bi+ cj+ dk : a, b, c, d ∈ R} ,where the imaginary units i, j, k satisfy i2= j2= k2= ijk=−1 and, conse-quently,ij = k = −ji, jk = i = −kj, ki = j = −ik,is an associative but noncommutative division algebra over R called quater-nionic skew field. The real and imaginary parts of a quaternion η = a+ bi+cj+dk are defined as Re η = a and Im η = bi+cj+dk, respectively, whereas,2similar to the complex case, the conjugate η is given by η = a− bi− cj− dkand the norm |η| is defined as |η| = √ηη.Two quaternions η and ν are said to be similar, η ∼ ν, if there exists anonzero α ∈ H such that η = ανα−1. Similarity is an equivalence relationand we denote by [ν] the equivalence class containing ν.Unlike the real or complex case, there are several possible ways to definequaternionic polynomials since the coefficients can be taken to be on theright, on the left or on both sides of the indeterminate (see, e.g., [9]). In thispaper we will adopt the following definition.The set of quaternionic polynomials is defined byH[s] ={p(s) =n∑l=0plsl, pl ∈ H, n ∈ N}.The sum and product of polynomials are defined as in the commutative casewith the additional rule (asn) (bsm) = absn+m.Conjugacy is extended to quaternionic polynomials by linearity and by therule asn = asn, ∀a ∈ H.3 Determinants of quaternionic and quater-nionic polynomial matricesBefore presenting the notion of determinant for quaternionic matrices wenote that, due to the noncommutativity of H, the definition adopted for realor complex matrices is not suitable in the quaternionic case.Indeed, let A ∈ Rn×n and denote by Al, l = 1, . . . , n the columns of A, i.e.,A =[A1| · · · |An]. It is well know that, by definition, the determinant of Asatisfies, among others, the following properties [2]i) det( [A1| · · · |αAl| · · · |An] )= α det( [A1| · · · |Al| · · · |An] ), α ∈ R;ii) det I = 1, where I is the identity matrix.Let, for instanceA =[i 00 j],3and suppose that the previous properties i) and ii) hold for quaternionicmatrices. ThendetA = det[i 00 j]= i det[1 00 j]= ij det[1 00 1]= ij = kwhereas, on the other handdetA = det[i 00 j]= j det[i 00 1]= ji det[1 00 1]= ji = −k,leading to an absurd. Therefore other defining properties must be consideredfor quaternionic determinants.After an unsuccessful attempt to define the determinant of a quaternionicmatrix made by Arthur Cayley in 1845 [3], only in the twentieth centurynew developments in this topic were achieved and some different definitionssuch as the determinants of Dieudonne´ [4], Study [10] and Moore [1] weregiven.Our definition of determinant for quaternionic polynomial matrices is basedon the definition of the Dieudonne´ determinant, with the necessary adjust-ments to the polynomial case.In order to introduce the Dieudonne´ determinant, we will use the followingnotation, according to [1].Denote by Plm the matrix that is obtained from the identity by interchangingthe lth and mth rows. Denote by Blm(α), where α ∈ C and C is a set, thematrix that is obtained from the identity by adding themth row multiplied byα to the lth row. Finally denote by SL(n, C) the set of all n×n matrices thatcan be decomposed as a product of matrices of the types Plm and Blm(α),α ∈ C.Definition 3. [4] Let A ∈ Hn×n; the Dieudonne´ determinant of A, denotedby Ddet(A), is defined as follows.• If A has not full rank, then Ddet(A) := 0.• Otherwise, let U ∈ SL(n,H) be such thatUA = diag(1, . . . , 1, α), α ∈ H. (2)Then Ddet(A) := |α|.4Note that Ddet can be regarded as a generalized determinant in the sensethat it satisfies the following properties:G1) It is zero if and only if the corresponding matrix has not full rank.G2) It satisfies the product rule.G3) It is equal to one for matrices of the type Blm(α).Definition 3 cannot be directly extended to the polynomial case, since it isin general not possible to bring up a polynomial matrix into the form (2).Therefore, in the definition that we propose, we consider instead a triangularform.Definition 4. We define the function Pdet(·) : Hn×n[s]→ R[s] as follows.Let R ∈ Hn×n[s]. Let further U ∈ SL(n,H[s]) be such that UR is uppertriangular, i.e.,UR = T =γ1 ∗ · · · ∗ ∗0 γ2. . .......0 0. . . ∗ ∗....... . . γn−1 ∗0 0 · · · 0 γn . (3)ThenPdet(R) :=n∏l=1γlγl.Example 5. LetR(s) =[(s+ 2j)(s+ j) (s+ 2j)(2s+ k)(s+ 3i) + 2s+ 3s+ j (2s+ k)(s+ 3i)].ThenR = UT =[s+ 2j 11 0] [s+ j (2s+ k)(s+ 3i)0 2s+ 3],and U ∈ SL(n,H[s]). ThereforePdet(R) = (s+ j)(s+ j)(2s+ 3)(2s+ 3) = (s2 + 1)(2s+ 3)2.5It turns out that the definition of the function Pdet(·) given above is well-posed, i.e., if there exists another triangular matrix T ′ 6= T , where T is thematrix defined in (3), such that T ′ = U ′R, with U ′ ∈ SL(n,H[s]), then theelements of the main diagonal of T ′, γ′1, . . . , γ′n, are such thatn∏l=1γ′lγ′l =n∏l=1γlγl.Moreover, it can be shown that Pdet(·) is a generalized determinant in thesense that it satisfies properties G1, G2 and G3.As should be expected, the zeros of Pdet(sI − A) can be related with theeigenvalues of the associated matrix A. A quaternion λ is said to be a righteigenvalue of A ∈ Hn×n if Av = vλ, for some nonzero quaternionic vectorv ∈ Hn. The vector v is called a right eigenvector associated with λ. The setσr(A) = {λ ∈ H : Av = vλ, for some v 6= 0}is called the right spectrum of A.Theorem 6. Let A ∈ Hn×n. Thenλ ∈ σr(A) ⇔ λ is a zero of Pdet(sI − A).This allows to relate the zeros of Pdet(R) with the right eigenvalues of thecompanion matrix [7] associated with R(s).Corollary 7. Let R(s) = Insm +Rm−1sm−1 + · · ·+R1s+R0 ∈ Hn×n[s] andA =0 In 0 · · · 00 0 In · · · 0.......... . ....0 0 0 · · · In−R0 −R1 −R2 · · · −Rm−1 ∈ Hmn×mnbe the block companion matrix of R. Thenλ ∈ σr(A) ⇔(PdetR)(λ) = 0.4 Quaternionic system stabilityThe definition of the polynomial determinant Pdet allows to extend Theorems1 and 2 on system stability to the quaternionic case.6Theorem 8. Let A ∈ Hn×n. Then the following statements are equivalent.(i) The quaternionic system described by x(t+ 1) = Ax(t) is stable.(ii) σr(A) ⊂ SZ := {q ∈ H : |q| < 1}(iii) All the zeros of Pdet(sI − A) lie in SZ.Based on this theorem and Corollary 7, it is not difficult to prove our finalresult.Theorem 9. Consider the systemR (σ)w = 0, (4)with R(s) := Rmsm + · · ·+R1s+R0 ∈ Hn×n[s]. This system is stable if andonly if all the zeros of Pdet(R(s))lie in SZ.AcknowledgementThe present research was partially supported by Unidade de Investigac¸a˜oMatema´tica e Aplicac¸o˜es, University of Aveiro, Portugal, through ProgramaOperacional “Cieˆncia, Tecnologia, Inovac¸a˜o” (POCTI) of the Fundac¸a˜o paraa Cieˆncia e Tecnologia (FCT), co-financed by the European Community fundFEDER.References[1] H. Aslaksen. Quaternionic determinants. Math. Intelligencer, 18(3):57–65, 1996.[2] E. T. Browne. Introduction to the theory of determinants and matrices.University of North Carolina Press, Chapel Hill, N. C., 1958.[3] A. Cayley. On certain results relating to quaternions. PhilosophicalMagazine, 26:141–145, 1845.[4] J. Dieudonne´. Les de´terminants sur un corps non commutatif. Bull.Soc. Math. Fr., 71:27–45, 1943.[5] D. Hestenes. New foundations for classical mechanics. Kluwer AcademicPublishers Group, Dordrecht, The Netherlands, 1986.7[6] T. Kailath. Linear Systems. Prentice Hall, Englewood Cliffs, 1980.[7] P. Lancaster and M. Tismenetsky. The theory of matrices. ComputerScience and Applied Mathematics. Academic Press Inc., Orlando, FL,1985.[8] J. W. Polderman and J. C. Willems. Introduction to Mathematical Sys-tems Theory: A Behavioral Approach, volume 26 of Texts in AppliedMathematics. Springer-Verlag, Berlin, 1997.[9] S. Pumplu¨n and S. Walcher. On the zeros of polynomials over quater-nions. Comm. Algebra, 30(8):4007–4018, 2002.[10] E. Study. Zur theorie der linearen gleichungen. Acta Math., 42:1–61,1920.[11] P. B. Visscher and X. Feng. Quaternion-based algorithm for micromag-netics. Phys. Rev. B, 65(104412), 2002.8

Stability of quaternionic systems: a determinantal approach

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