Computation of the Smith Form for Multivariate Polynomial Matrices Using Maple

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1 American Journal of Comutational Mathematics,,, -6 htt://dxdoiorg/436/ajcm3 Published Online March (htt://wwwscirporg/journal/ajcm) Comutation of the Smith Form for Multivariate Polynomial Matrices Using Male Mohamed Salah Boudellioua Deartment of Mathematics and Statistics, Sultan Qaboos University, Muscat, Oman Received October 3, ; revised December, ; acceted December 5, ABSTRACT In this aer we show how the transformations associated with the reduction to the Smith form of some classes of multivariate olynomial matrices are comuted Using a Male imlementation of a constructive version of the Quillen-Suslin Theorem, we resent two algorithms for the reduction to a articular Smith form often associated with the simlification of linear systems of multidimensional equations Keywords: Smith Form; Unimodular; Equivalence; Quillen-Suslin Theorem; Male Introduction Matrices whose elements are olynomials in more than one indeterminate have been studied by many authors Such matrices arise in the mathematical treatment of the so-called multidimensional systems which can be considered as extensions of the ordinary differential or difference systems These include delay-differential systems and artial differential systems In articular the Smith normal form of a matrix lays an imortant role in many areas of mathematics such as the olynomial aroach in control theory see for examle Rosenbrock [] and Kailath [] The roblem of reducing an univariate olynomial matrix to its Smith form is well understood and the relevant algorithm is already imlemented in most comuter algebra systems For the multivariate case, however the roblem is still oen Desite that some necessary and/or sufficient conditions for the reduction of a matrix to its Smith form have been given in the literature, no algorithm has been given to show how the transformations involved in the reduction are actually comuted These transformations are imortant if the solutions of the reduced system are to be exressed in terms of the original variables So far the comutations associated with these conditions have been difficult if not imossible to carry out Recently however, some rogress has been made in symbolic comutation in articular a QuillenSuslin Male ackage [3] has been develoed This ackage which is a male imlementa- tion of the Quillen-Suslin theorem rovides an algorithm which comutes a basis of a free module over a olynomial ring In terms of matrices, this algorithm comletes a unimodular rectangular matrix to an invertible matrix over the given olynomial ring with rational or integer coefficients (for more details see htt://wwwbmath rwth-aachende/quillensuslin/) In this aer, we show how this ackage can be used to comute the Smith form and the associated transformation for some classes of multivariate olynomial matrices The classes of matrices considered can be regarded as those associated with linear determined systems of multidimensional equations which can be reduced to a single equation, thereby simlifying the analysis of such systems The transformation used to obtain the Smith form is that of unimodular equivalence which will be defined later First we need to introduce some definitions and a theorem which lay a key role in this aer Definition Let D be a ring The general linear grou GL D is defined by = : = = GL D M D N D MN NM I An element M GL D matrix In the case where x x is called a unimodular D= K,, n, a olynomial ring in the indeterminates x,, xn and coefficients in K, the matrix M is unimodular if and only if the determinant of M is invertible in D, ie, is a non-zero element of K In the case when a matrix is rectangular we introduce the following concet of rimeness q Definition A matrix R D with > q is said to be zero-left-rime ZLP if it admits a right-inverse q over D ie, if there exists R D such tha t RR = I q Similarly zero-right-rimeness ZRP can be defined by transosition for matrices w here < q We now state the famous Quillen-Suslin Theorem Coyright SciRes

2 M S BOUDELLIOUA This theorem is at the centre of the roofs of the results resented in this aer and its Male imlementation is used in the comutation of the transformations Theorem (Quillen-Suslin [4,5]) Let K be a rincial ideal domain and D K x,, x n and let q R D be a matrix which admits a right-inverse q R D, ie, RR Iq Then there exists a unimodular matrix NGL D such that RN Iq () Equivalence to the Smith Form The Smith form S of a q matrix T with elements in a ring D is usually the result of an equivalence transformation, ie a transformation of the form S MTN () and where M GL D M GLq D The resulting Smith f orm S is given by the diagonal matrix formed by the invariant olynomials The non-zero invariant olynomials,, r are defined by i i i with, r is the rank of T and i is the gcd of the i i minors of T It is a fact that it is not always ossible to reduce a matrix to its Smith form by an equivalence of the tye () In order to show that any matrix can be brought by an equivalence transformation to its Smith form, it is usually required that D is a rincial ideal or a Euclidean domain In the case when D z,, zn, the matrices may be treated as having elements in one of the indeterminates with coefficients that are rational forms in the other indeterminates eg z,, zn zn This aroach which involves a renormalization ste has the disadvantage of yielding Smith forms which are not unique, see the work of Frost and Storey [6] and Morf et al [7] Conditions under which a matrix with elements in z,, zn is equivalent to its Smith form have been given by a number of authors For the ring z, z, Lee and Zak [8] roosed a necessary and sufficient condition in terms of the existence of solutions to certain olynomial equations Frost and Boudellioua [9] gave a necessary and sufficient condition for a class of multivariate olynomial matrices in terms of the existence of a olynomial vector Lin et al [] roosed a sufficient condition for a class of matrices whose determinant is linear in one of the indeterminates Theorem (Boudellioua and Quadrat []) Let,, D z zn, a matrix T D is equivalent to the Smith form: S I T if and only if there exists a ZRP vector U D (3) such that the matrix T U is ZLP This is a articular tye of Smith form where all the invariant olynomials are equal to excet the last one which is given by the determinant of the matrix This form is imortant for simlification considerations, ie a system whose matrix is equivalent to the Smith form (3) is equivalent to a single equation in one unknown Thus making it easier to analyse such a system either analyticcally or numerically In order to exress the conclusions and solutions made about the reduced system in terms of the original system, one has to comute the transformations () connecting the original system matrix to the Smith form Suose that such a vector is obtained then the transformations M and N that reduce T to the Smith form (3) can be obtained via the followin g Male algorithm Algorithm Declare the ath where the required Male libraries such as QuillenSuslin are stored and load the ackages LinearAlgebra and QuillenSuslin Declare the ring over which the matrix is defined by declaring the indeterminates and the field of coefficients Enter the elements of the matrices T and U Test the zero-rimeness of the matrix T U using the function IsUnimod Use the function QSAlgorithm to comute the square T T unimodular matrix M such that U M I T Interchange the first and last colu mn of M and transose to obtain the matrix M Extract the matrices Ki, i 4 and Lj, j from M and T resectively Comute the matrix T KL KL, where K K L M and T K3 K4 L Use the function QSAlgorithm to obtain the square unimodular matrix N such that TN I Extract the submatrix R formed from the first row and the first columns of the matrix K3L K4L K K Construct the matrix M = RKK3 RK K4 Check that the Smith for m S = MTN Another interesting result in the form of a sufficient condition for the reduction of class of linear multidimensional systems was given by Lin et al [] This is the class of systems where the determinant of the system matrix is linear in one of the indeterminates Such systems are also equivalent to a single equation The result is given for the case when the determinant is linear in z but it is equally valid for any indeterminate zi, i =,, n Theorem 3 (Lin et al []) Let D = z,,z n Coyright SciRes

3 M S BOUDELLIOUA 3 and T D, then if T = z f z,, z n, T is eq- uivalent to the Smith form: I S T In the following we give an algorithm to find the unimodular matrices M and N that reduce the matrix T to the Smith form S Algorithm Declare the ath where the required Male libraries such as QuillenSuslin are stored and load the ackages LinearAlgebra and QuillenSuslin Declare the ring D over which the matrix is defined by declaring the indeterminates and the field of coefficients Enter the elements of the matrix T Check that the determinant of T is linear i n one of the indeterminates, say z i Comute f = z i T Substitute zi = f in T to obtain T Comute a ZRP vector w, zi, z, i such that Tw = using the function SyzygyModule Comute a matrix N GL D with last column given by w using the function ComleteMatrix M GL D given by Comute the matrix I M = SN T, where S T Check that the Smith form S = MTN Examle (Frost and Boudellioua [9]) libname:= C:/Involutive, C:/QuillenSuslin, libname: with (LinearAlgebra): with (QuillenSuslin): Consider again the 3 3 matrix T in Examle over the olynomial ring D= s,z var := [s, z]; var : s, z T := Matrix([[*s*z^ + z^3 + z^ +,s*z^ s*z + s, * s*z + z^], [*s*z + z^ + z, s*z s, *s + z], [*s^*z + s*z^ + s*z + z, s^*z s^, *s^ + s*z + ]]); det_t := Determinant(T); 3 sz z z sz szs szz T : szz z szs sz s zsz szz s zs s sz := RowDimension(T); Now consider the column vector U satisfying the condition in Theorem 3, det_ T : s z : 3 (4) U := Matrix([[z], [], [s]]); UT := Transose(U); z U : s UT = z IsUnimod(UT, var, ); TU := Matrix([T,U]); TU : 3 sz z z sz szs szz z szz z szs sz s zsz szz s zs s sz s Let us check if the the matrix IsUnimod (TU, var, ); s is unimodular Alying the QSAlgorithm rocedure to the row UT, we then obtain MT := QSAlgorithm(UT, var, ); T U MT : z s UTMT := simlify(matrix(ut) MT); UTMT : Now we can check that UT is the first row of the inverse of MT : MT_inv := ComleteMatrix(UT, var, ); z s MT _ inv : ie, the matrix MT_ inv is a comletion of UT to an invertible matrix over D M := Transose(ColumnOeration (Matrix(MT), [, ] )); s M: z K K The matrix M K3 K 4 and T L L T, where K : = DeleteColumn(DeleteRow(M, ), ); K := DeleteColumn(DeleteRow(M, ), ); K3 := Coyright SciRes

4 4 M S BOUDELLIOUA DeleteColumn(DeleteRow(M, ), ); K4 := DeleteColumn(DeleteRow(M, ), ); K: K : K3: K 4: s z L := DeleteRow(T, ); L : = DeleteRow(T, ) ; 3 sz z z sz szs szz L: szz z szs sz L s zsz szz s zs s sz : T := simlify(kl + KL); z T: s N := QSAlgorithm(T, var, ); s N : z sz R := simlify(submatrix((k3l + K4L) N,, )); M := Matrix([[K, K], [ RK + K3, RK + K4]]); S := simlify(mtn); R: s z z s M: z z szsz sz S: s z Examle (Frost and Boudellioua [9]) libname := C: /Involutive, C: /QuillenSuslin, libname: with (LinearAlgebra): with (QuillenSuslin): In the QuillenSuslin ackage all comutations are erformed over a commutative olynomial ring with rational coefficients if the last arameter aram is set to and with integer coefficients, if the arameter is set to false aram := : The variables s and z of the olynomial ring D= s, z mus t be declared by setting: var := [s,z]; var : [ s, z] Now consider the 3 3 matrix T(, s z ) over the olynomial ring D T := Matri x( [[*s*z^ + z^3 + z^ +, s*z^-s*z + s, * s*z + z^],\ [*s^*z + s*z^ + s*z + z, s^*z s^, *s^ + s*z + ]]); det_t := Determinant(T); 3 sz z z sz szs szz T : szz z szs sz s z sz szz s zs s sz det_ T : s z Clearly the determinant of dition in Theorem 3 := RowDimension(T); f := s-det_ T; : 3 f : z T(, s z) satisfies the con- Evaluating T(, s z ) at s = z, ie comuting Tz, z P z, P := subs(s = f, T); 3 3 z z z z z z P: z z z z z 3 3 z z z z z z To find a unimodular column vector w satisfying Pw = we need to find the row vector defining the syzygy module of P over D w := Transose(Matrix(SyzygyModule(Transose(P), var))); z w : z Testing that the vector w is unimodular and that Pw IsUnimod (w,var,aram); Now comute a matrix NGL3 lumn given by w z with last co- N := ColumnOeration (Transose(ComleteMatrix (Transose(w),\ var,false)),[,]); PN := simlify(pn); z N : z Coyright SciRes

5 M S BOUDELLIOUA 5 Comuting the matrix I MB 3 z z z PN : z z z 3 z z z z 3 N GL z such that M := Transose (QSAlgorithm(Transose(B), var, )); z z z ( z z ) z z z : z z z 3 M z z z TN := simlify(tn); 3 3 szz sz z z z zsz szsz s sz s zsz szz sz s zs sz TN : s z sz z z z sz s z S := Matrix([[IdentityMatrix( ), ZeroMatrix(,)], \[ZeroMatrix(, ), det_t]]), S : s z M := simlify(smatrixinverse(n) MatrixInverse(T)); N = N; M : 3 z z s zsz z z zs z sz z s szz z sz z s szz sz IsUnimod(M,var,); SS := simlify(mtn); 3 Conclusion z N z SS : s z In this aer, we have shown that the recent imle- of a constructive version of the Quillen-Suslin Theorem can be used effectively to com- mentation in Male ute the Smith form and the associated unimodular transformations for a class of multivariate olynomial matrices The classes of matrices considered are those arising from multidimensional systems amenable to be simlified to a single equation in one unknown The case of underdetermined systems can also be treated in a similar fashion 4 Acknowledgements The author wishes to exress his thanks to Sultan Qaboos University (Oman) for their suort in carrying out this research and Dr Alban Quadrat and Anna Fabianska for their hel in roviding the author with a coy of the QuillenSuslin Male ackage REFERENCES [] H H Rosenbrock, State Sace and Multivariable Theory, Nelson-Wiley, London, 97 [] T Kailath, Linear Systems, Prentice-Hall, Englewood Cliffs, 98 [3] A Fabianska and A Quadrat, Alications of the Quillen-Suslin Theorem to Multidimensional Systems Theory, Technical Reort 66, INRIA, Sohia Antiolis, 7 [4] D Quillen, Projective Modules over Polynomial Rings, Inventiones Mathematicae, Vol 36, 976, 67-7 doi:7/bf398 [5] A Suslin, Projective Modules over Polynomial Rings Are Free, Soviet Mathematics Doklady, Vol 7, No 4, 976, 6-64 [6] M Frost and C Storey, Equivalence of a Matrix over R[s; z] with Its Smith Form, International Journal of Control, Vol 8, No 5, 979, doi:8/ [7] M Morf, B Levy and S Kung, New Results in -D Systems Theory: Part I: -D Polynomial Matrices, Factorization and Corimeness, Proceedings of the IEEE, Vol 65, No 6, 977, doi:9/proc97758 [8] E Lee and S Zak, Smith Forms over R[z; z], IEEE Transactions on Automatic Control, Vol 8, No, 983, 5-8 doi:9/tac98338 [9] M Frost and M S Boudellioua, Some Further Results Concerning Matrices with Elements in a Polynomial Ring, International Journal of Control, Vol 43, No 5, 986, doi:8/ [] Z Lin, M S Boudellioua and L Xu, On the Equivalence and Factorization of Multivariate Polynomial Ma- Coyright SciRes

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