To appear in International Journal of Nonlinear Analysis and Applications Vol. 00, No. 00, Month 20XX, 1 10 PAPER

Transcription

1 May, 06 International Journal of Nonlinear nalysis and pplications Petraki&Samaras To appear in International Journal of Nonlinear nalysis and pplications Vol 00, No 00, Month 0XX, 0 PPER Solving the nth Degree Polynomial Matrix Equation N Samaras a and D Petraki a a Department of pplied Informatics, School of Information Sciences, University of Macedonia, 56 Egnatia Str, Thessaloniki, Greece (released October 03 The algorithm for finding the nth roots of a matrix is well-known The aim of this paper is to present the general case of the nth degree polynomial matrix equation The study of the general case help us to solve any polynomial matrix equation The main difficulty to solve the polynomial matrix equation is that, in general, the nth degree polynomial function h(x, is not invertible Even if the function h is invertible, it is difficult to find the type of the inverse function and its derivatives We designed an algorithm, which enables us to bypass anything related with the inverse function of h In our algorithm we just used the polynomial function h and its derivatives This is a very effective procedure and our algorithm can be used for every polynomial function h and any square matrix ll the possible cases concerning the Jordan canonical form of the matrix are examined Mathematical types to calculate the number of different roots of the polynomial matrix equation and their algebraic multiplicity are also presented Keywords: Polynomial matrix equation, Simple Matrix, Derogatory Matrix, Type of roots, Interpolating polynomial MS Subject Classification: Cxx, 5xx, 65H04 Introduction Matrix theory was developed by ugustin Cauchy ( , rthur Cayley (8-895, James Sylvester (84-897, Ferdinard Frobenius (849-97, Leopold Kronecker (83-89, Karl Theodor Weierstrass ( and others In 858 Cayley in his seminar Memoir on the Theory of Matrices, investigated the square root of a matrix Sylverster proposed definitions of f( for general f There are four equivalent definitions of f(, based on the Jordan canonical form, polynomial interpolation, the components of a matrix and the Cauchy integral formulae One of the earliest uses (938 of matrix theory in practical applications was by Robert Frazer, William Duncan, and rthur Collar of the erodynamics Department of the National Physical Laboratory of England, who were developing matrix methods for analyzing unwanted vibrations in aircraft Corresponding author samaras@uomgr; Tel: ; Fax:

2 May, 06 International Journal of Nonlinear nalysis and pplications Petraki&Samaras Consider the following nth degree polynomial matrix equation: h(x = a n X n + +a X + a 0 I ν =, ( where, XϵM ν ν (K, K = R or K = C The problem of determining all the nth roots X of a given matrix has been examined by many mathematicians [-8 Bjorck and Hammarling [9 show how to compute a cubic root using the complex Schur decomposition n algebraic formulae giving the square roots of matrices is presented in [0 study of the matrix approach to polynomials and its further exploration is presented in [, In [3, the determination of algebraic formulae giving all the solutions of the matrix equation X n =, where n > and is a matrix with real or complex elements is presented Some other methods for computing the nth root are described by Bini et al in [4 In this paper an algorithm is presented for the first time, for solving any polynomial matrix equation of the form (, with the use only of the polynomial function h(x and its derivatives Furthermore, we compute the number and the type of the roots of a polynomial matrix equation and also their algebraic multiplicities Basic Properties In this section we describe five basic propositions related to matrices and polynomial equations Proposition If M is any arbitrary matrix ν ν, J M is his Jordan canonical form, S M is the transition matrix and f(x is a polynomial, then f(m = S M f(j M S M Proof See theorems 94, 94, 943 in [5 or in [3 for a complete proof Re[λ Im[λ Proposition If λϵc and M =, where Re[λ and Im[λ Im[λ Re[λ are the( real and imaginary part of λ, then M k Re[λ = k Im[λ k Im[λ k Re[λ k, kϵn Proof The ( proof is obvious using the method of induction for k It is, M Re[λ Im[λ = true Im[λ Re[λ ( Let M k Re[λ = k Im[λ k Im[λ k Re[λ k, then ( M k+ = M k M Re[λ = k Re[λ Im[λ k Im[λ Re[λ k Im[λ + Im[λ k Re[λ (Re[λ k Im[λ + Im[λ k Re[λ Re[λ k Re[λ Im[λ k Im[λ ( Re[λ = k+ Im[λ k+ Im[λ k+ Re[λ k+

3 May, 06 International Journal of Nonlinear nalysis and pplications Petraki&Samaras Re[λ Im[λ Proposition 3 If λϵc, M = and f is a polynomial with real Im[λ Re[λ Re[f(λ Im[f(λ coefficients then f(m = Im[f(λ Re[f(λ Proof It is an obvious conclusion from proposition 3 Proposition 4 If q(x is a polynomial of ν degree, h(x is a polynomial of (k n degree which satisfies the relations, q (k (λ =, k =,,, ν h ( (q(x then, (hoq ( (x = and (hoq (k (x = 0, k =, 3,, ν Proof pplying the derivatives rules for function hoq(x the following results are obtained (hoq ( (x = (h(q(x ( = h ( (q(xq ( (x = h ( (q(x h ( (q(x = (hoq ( (x = (h(q(x ( = h ( (q(xq ( (x + h ( (q(xq ( (x = = h( (q(x (h ( (q(x + h ( (q(xq ( (x = h( (q(x (h ( (q(x + h ( (q(x h( (q(xq ( (x (h ( (q(x = h( (q(x (h ( (q(x h( (q(xh ( (q(x (h ( (q(x h ( (q(x = h( (q(x (h ( (q(x h( (q(x (h ( (q(x = 0 So (hoq (k = 0, k = 3, 4,, ν Proposition 5 If q(x is a polynomial of ν degree and h(x is a polynomial of n degree which satisfies the relations (k h(ρ = λ, q(λ = ρ, q (k (λ =, k =,,, ν then h ( (q(x x=λ x=λ (hoq(λ = λ, (hoq ( (λ = and (hoq (k (λ = 0, k =, 3,, ν Proof It is an obvious conclusion from proposition 34 3 Solving the nth degree polynomial matrix equation 3 Calculation of the roots Let a n X n + a n X n + + a X + a 0 I ν =, with, XϵM ν ν (K, K = R or K = C be a nth degree polynomial matrix equation and h (x = a n x n + a n x n + + a x + a 0, be the corresponding polynomial There are two basic cases concerning the algebraic multiplicity of the eigenvalues of the matrix Case The matrix is simple If ρ, ρ,, ρ ν be numbers such that h(ρ i = λ i, i =,, 3,, ν and q(x be the interpolating polynomial to the data (λ i, ρ i, i =,, 3,, ν then (hoq( = Proof If = S J S then it is (hoq( = (hoq(s J S = S (hoq(j S = S diag[(hoq(λ, (hoq(λ,, (hoq(λ ν S = S diag[h(q(λ, h(q(λ,, h(q(λ ν S 3

4 May, 06 International Journal of Nonlinear nalysis and pplications Petraki&Samaras = S diag[h(ρ, h(ρ,, h(ρ ν S = S diag[λ, λ,, λ ν S = S J S = Corollary 3 The matrix B = q( is a root of the polynomial matrix equation a n X n + a n X n + + a X + a 0 I ν = Case The matrix is derogatory If λ, λ,, λ s be the, different by two, eigenvalues of matrix with algebraic multiplicities α, α,, α s, geometric multiplicities γ, γ,, γ s and indices d, d,, d s, respectively Let be ρ i, i =,,, s numbers such that h(ρ i = λ i, i =,,, s and q(x be the ( interpolating ( polynomial to the data ( ( (αi λ i, ρ i, h ( (q(x, x=λ h ( (q(x, i x=λ h ( (q(x,, i x=λ h ( (q(x, i x=λ i i =,, 3,, s then (hoq( = Proof The Jordan canonical form of matrix is J = diag[λ I γ, J d,, λ i I γi, J di,, λ s I γs, J ds, where J di, i =,,, s, are matrices λ i λ i 0 0 d i d i of the form J di = 0 0 λ i [ λ i It is (hoq( = (hoq(s J S = S (hoq(j S = = S diag[(hoq(λ I γ, (hoq(j d,, (hoq(λ i I γi, (hoq(j di,, (hoq(λ s I γs, (hoq(j ds S = S diag[λ I γ, J d,, λ i I γi, J di,, λ s I γs, J ds S = (from 34, 35, so (hoq( = or h(q( = Corollary 3 Let be B = q( then h(b =, so the matrix B = q( is a root of the polynomial matrix equation a n X n + a n X n + + a X + a 0 I ν = 3 Comment If we work in the real vector space M ν ν (R and the matrix has as eigenvalues ( the conjugate complex numbers λ and λ then the corresponding Jordan block is Re[λ Im[λ Im[λ Re[λ Proof ( We can assume, without restricting the generality, that Re[λ Im[λ J = Im[λ Re[λ There is the invertible transition ν ν matrix S such that = S J S Let ρ be a complex number such that h(ρ = λ and q(x be the first degree interpolating polynomial, with real coefficients, fitted to the data (λ, ρ It is (hoq( = (hoq(s J S = S (hoq(j S, Re[(hoq(λ Im[(hoq(λ with (hoq(j = (from 3, 33, Im[(hoq(λ Re[(hoq(λ Re[λ Im[λ so (hoq( = = J Im[λ Re[λ 4

5 May, 06 International Journal of Nonlinear nalysis and pplications Petraki&Samaras It is (hoq( = S J S = or h(q( = Let be B = q( then h(b =, so the matrix B = q( is a root of the polynomial matrix equation a n X n + a n X n + + a X + a 0 I ν = 4 The number of the roots of nth degree polynomial matrix equation and their algebraic multiplicities There are four main cases concerning the algebraic multiplicity of the eigenvalues of the matrix 4 Case ll the eigenvalues λ, λ,, λ ν of the matrix have algebraic multiplicity and each of the equations h(x = λ i, i =,, 3,, ν has n different by two roots In this case the equation a n X n +a n X n ++a X+a 0 I ν = has n ν different roots Proof There are ν different equations h(x = λ i, i =,, 3,, ν with n different by two roots each of them, so by the fundamental rule of counting there are m = n n n = n ν different ways, so the equation has n ν different roots n illustrative example Let be = We will solve the polynomial matrix equation X 5X + 7I = Let be h(x = x 5x + 7 It is n = (degree of the polynomial h(x and ν = (dimension of the matrix The characteristic polynomial of matrix is p(x = x 4x + 3 The eigenvalues of the matrix are λ =, λ = 3 with algebraic multiplicities a = and a = respectively The equation h(x = λ has roots ρ = and ρ = 3 with algebraic multiplicities α = and α = respectively The equation h(x = λ has roots ρ = and ρ = 4 with algebraic multiplicities α = and α = respectively So the equation has n ν = 4 different by two roots X[i with algebraic multiplicities, for i =,, 3, 4 The matrix of the interpolation data is The roots of the equation are ( λ = ρ = ρ = 3 λ = 3 ρ = ρ = 4 ( The interpolating polynomial for which is q[( = ρ =, q[(3 = ρ = is q[(x = x + 5, and the correspondent matrix root is X[ = q[( = ( 3 3 ( The interpolating polynomial for which is q[( = ρ =, q[(3 = ρ = 4 is q[(x = x +, and the correspondent matrix root is 3 X[ = q[( = 3 (3 The interpolating polynomial for which is q[3( = ρ = 3, q[3(3 = ρ = is q[3(x = x + 4, and the correspondent matrix root is X[3 = q[3( = 5

6 May, 06 International Journal of Nonlinear nalysis and pplications Petraki&Samaras (4 The interpolating polynomial for which is q[4( = ρ = 3, q[4(3 = ρ = 4 is q[4(x = x + 5, and the correspondent matrix root is X[4 = q[4( = ( Case ll the eigenvalues λ, λ,, λ ν of the matrix have algebraic multiplicity and each of the equations h(x = λ i has m i different roots, ρ i, ρ i,, ρ imi and each of them has algebraic multiplicity α i, α i,, α imi, where i =,, 3,, ν In this case the number of different roots of the polynomial matrix equation is m = m m m ν Each matrix root of the above m has as algebraic multiplicity the product of the algebraic multiplicities of the ordinates corresponding to the interpolating data This means that if {(λ, ρ j, (λ, ρ j,, (λ ν, ρ νjν } is a set of interpolating data then the algebraic multiplicity of the produced matrix root X is α i α i α imi Proof There are ν different equations h(x = λ i, i =,, 3,, ν with m i different by two roots each of them, so by the fundamental rule of counting there are m = m m m ν different ways, so the equation has m different roots n illustrative example 4 3 Let be = We will solve the polynomial matrix equation X 3 3X + 5I = Let be h(x = x 3 3x + 5, then h ( (x = 3x 6x, h ( (x = 6x The characteristic polynomial of matrix is p(x = x 6x + 5 The eigenvalues of the matrix are λ =, λ = 5 with algebraic multiplicities a = and a = respectively The equation h(x = λ has m = different roots ρ = and ρ = with algebraic multiplicities α = and α = respectively The equation h(x = λ has m = different roots ρ = 3 and ρ = 0 with algebraic multiplicities α = and α = respectively So the equation has n ν = 3 = 9 roots The number of the different roots is m = m m = = 4 λ = ρ The matrix of the interpolation data is = ρ = λ = 5 ρ = 3 ρ = 0 The roots of the equation are ( The interpolating polynomial for which is q[( = ρ =, q[(5 = ρ = 0 is q[(x = x 4 5 4, and the correspondent matrix root is X[ = q[( = ( The interpolating polynomial for which is q[( = ρ =, q[(5 = ρ = 3 is q[(x = x, and the correspondent matrix root is 3 X[ = q[( = 0 (3 The interpolating polynomial for which is q[3( = ρ =, q[3(5 = ρ = 0 is q[3(x = x + 5, and the correspondent matrix root is X[3 = q[3( = ( 3 3 6

7 May, 06 International Journal of Nonlinear nalysis and pplications Petraki&Samaras (4 The interpolating polynomial for which is q[4( = ρ =, q[4(5 = ρ = 3 is q[4(x = x , and the correspondent matrix root is X[4 = q[4( = ( Case 3 There exists at least one eigenvalue λ i of the matrix with algebraic multiplicity greater than In this case we do not know the exact number of the roots of the equation Let the matrix has s different eigenvalues λ, λ,, λ s, where s < ν, with algebraic multiplicities α, α,, α s respectively If the equation h(x = λ i has m i different roots ρ i, ρ i,, ρ imi with algebraic multiplicity α i, α i,, α imi respectively, i =,,, s, then the number of different roots of the polynomial matrix equation is at least m = m m m s Proof See Case n illustrative example 4 0 Let be = We will solve the polynomial matrix equation X 0 4 = Let be h(x = x It is n = (degree of the polynomial h(x and ν = (dimension of the matrix The characteristic polynomial of matrix is p(x = (x 4 The matrix has only one eigenvalue λ = 4 with algebraic multiplicity a = The equation h(x = λ has the roots ρ = and ρ = with algebraic multiplicities α = and α = respectively and m = So the given equation has an undefined number of roots Let be h (x = h ( (x then the matrix of the interpolation data is ( λ = 4 ρ = h (ρ = 4 λ = 4 ρ = h (ρ = 4 The roots of the equation are ( The interpolating polynomial for which is q[(4 = ρ =, q ( [(4 = h (ρ = 4 is q[(x = x 4 +, and the correspondent matrix root is X[ = q[( = 0 0 ( The interpolating polynomial for which is q[(4 = ρ =, q ( [(4 = h (ρ = 4 is q[(x = x 4, and the correspondent matrix 0 root is X[ = q[( = 0 It can be verified that the matrix equation X = has as roots also the following ( matrices: a a,, aϵr 0 0 So the equation has an infinity number of roots n illustrative example 7

8 May, 06 International Journal of Nonlinear nalysis and pplications Petraki&Samaras Let be = We will solve the polynomial matrix equation X + X + 3I 4 = Let be h(x = x + x + 3, then h ( (x = x +, h ( (x = The characteristic polynomial of matrix is p(x = (x (x 3 3 The eigenvalue λ = has algebraic multiplicity a = The eigenvalue λ = 3 has algebraic multiplicity a = 3, so the given equation has an undefined number of roots The equation h(x = λ has m = root, ρ = with algebraic multiplicity α = The equation h(x = λ has m = roots, ρ = 0 and ρ = The matrix of the interpolation data is λ = ρ = λ = 3 ρ = 0 h (ρ = h( (ρ = λ = 3 ρ = h (ρ = h( (ρ = Two roots of the given equation are ( The interpolating polynomial for which is q[( = ρ =, q[(3 = ρ = 0, q ( [(3 = h (ρ =, q( [(3 = h ( (ρ = is q[(x = x3 and the correspondent matrix root is X[ = q[( = x + 35x ( The interpolating polynomial for which is q[( = ρ =, q[(3 = ρ =, q ( [(3 = h (ρ =, q( [(3 = h ( (ρ = is q[(x = 3x x 77x and the correspondent matrix root is X[ = q[( = Perhaps the given equation has and other matrix roots 44 Case 4 There exists at least one eigenvalue λ of the matrix with algebraic multiplicity greater than or equal to two, for which the equation h(x = λ has as root ρ the eigenvalue λ with algebraic multiplicity greater than or equal to two also, and then the algorithm can be applied The polynomial matrix equation is impossible or has an unknown number of roots Proof It is h ( (ρ = 0, so the corresponding interpolation data are not exist 8

9 May, 06 International Journal of Nonlinear nalysis and pplications Petraki&Samaras n illustrative example 0 Let be = We want to solve the polynomial matrix equation X 0 0 = Let be h(x = x It is n = (degree of the polynomial h(x and ν = (dimension of the matrix The characteristic polynomial of matrix is p(x = x The matrix has one eigenvalue λ = 0 with algebraic multiplicities a = The equation h(x = λ has the root ρ = 0 with algebraic multiplicity α = So our algorithm cannot be applied, and we must try to examine if our equation has a solution It is easy to verify that the equation does not have a solution, therefore it is impossible n illustrative example Let be = We want to solve the polynomial matrix equation X = Let be h(x = x It is n = (degree of the polynomial h(x and ν = 3 (dimension of the matrix The characteristic polynomial of matrix is p(x = x (x 4 The matrix has as eigenvalues the numbers λ = 0 with algebraic multiplicities a = and λ = 4 with algebraic multiplicities a = The equation h(x = λ has the root ρ = 0 with algebraic multiplicity α = The equation h(x = λ has the root ρ = 4 with algebraic multiplicity α = Hence, our algorithm cannot be applied, and we must try to examine if our equation has a solution It is easy to verify that the matrices ± a, ± are the roots of the a 0 polynomial matrix equation X =, aϵr 5 lgorithm Our paper is completed by presenting the algorithm that occurs from the previous examples, in cases where the equation has a finite number of roots Step Calculation of the different by two eigenvalues λ, λ,, λ s and their algebraic multiplicities α, α,, α s Let k is set to be the biggest of the above algebraic multiplicities Step The function h (x = up to order k h ( (x is defined and its derivatives are calculated Step 3 Solution of the equations h(x = λ i, for i =,,, s and their different roots are symbolized with ρ ij, i =,,, s and j =,,, m i Step [( 4 The ( definition of the interpolating data d = T able λ i, h (ρ iji, h ( (ρ iji,, h (α (ρ iji, i =,,, s, j =,,, m i We find the corresponding interpolating polynomials q(x Step 5 Matrices X = q( are solutions of the given polynomial matrix equation 9

10 May, 06 International Journal of Nonlinear nalysis and pplications Petraki&Samaras 6 Conclusion In this paper the algorithm for solving the nth degree polynomial matrix equation is developed Formulae are created in order to calculate the number of the equation s roots Furthermore, the cases where the equation has no roots or has an infinite number of roots are presented The results obtained from this work are the necessary and sufficient tools to solve and study the nth degree polynomial matrix equation References [ N J Higham, Functions of Matrices, Theory and Computation, Society of Industrial and pplied Mathematics, Philadelphia(US, 008 [ B Iannazzo, On the Newton method for the matrix n-th root, SIM J Matrix nal ppl 8: (006, pp [3 P Psarrakos, On the n-th roots of a compex matrix, Electron J Linear lgebra 9 (00, pp3-4 [4 C Guo and N J Higham Schur-Newton method for the matrix pth root and its inverse SIM J Matrix nal ppl 998; 8:3, [5 S Lacic, On the computation of the matrix k-th root, Z ngew Math Mech 78:3 (998, pp67-7 [6 N J Higham, Computing real square roots of a real matrix, Linear lgebra ppl 88/89 (987, pp [7 G lefeld and N Schneider, On square roots of M-matrices, Linear lgebra ppl 4 (98, pp9-3 [8 G Cross and P Lancaster, Square roots of complex matrices, Linear and Multilinear lgebra ( 974, pp89-93 [9 Bjorck and S Hammarling, Schur method for the square root of a Matrix, Linear lgebra ppl5/53 (005, pp [0 D Sullivan, The square roots of x matrices, Math Mag66 (993, pp34-36 [ T rponen, Matrix approach to polynomials, Linear lgebra ppl (004pp394, [ T rponen, Matrix approach to polynomials, Linear lgebra ppl 359 (004, pp8-96 [3 Choudhry, Extraction of nth roots of x matrices, Linear lgebra ppl387 (004, pp83-9 [4 D Bini, N J Higham and B Beni, lgorithms for the matrix pth root, NumerI lgorithms39:4 (005, pp [5 P Lancaster and M Tismenetsk, Theory of matrices California, cademic Press, San Diego(US,985 0