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Jordan Canonical Form with Parameters from Frobenius Form with Parameters Jordan Canonical Form with Parameters from Frobenius Form with Parameters Robert M. Corless, Marc Moreno Maza, and Steven E. Thornton ORCCA, University of Western Ontario, London, Ontario, Canada Motivation Parametric Matrix Computations in CAS Computer algebra systems generally give generic answers for computations on matrices with parameters. 1 Example Jordan canonical form of:2 3 6 α 0 1 7 6 7 6 7 6 0 α − 3 0 7 with α 2 C 4 5 0 0 −2 Maple Mathematica SymPy Sage 2 3 2 3 2 3 2 3 6 −2 7 6 −2 7 6 α − 3 7 6 7 6 −2 7 6 7 6 7 6 7 6 7 6 7 6 7 6 α 7 6 7 6 α 7 6 α 7 4 5 6 −3 + α 7 4 5 4 5 4 5 α − 3 α − 3 −2 α 2 Example Jordan canonical form2 of: 3 2 3 6 α 0 1 7 6 −2 0 1 7 6 7 6 7 6 7 α = −2 6 7 6 0 α − 3 0 7 −−−−! 6 0 −5 0 7 4 5 4 5 0 0 −2 0 0 −2 Maple Mathematica SymPy Sage 2 3 2 3 2 3 2 3 6 −5 7 6 −5 7 6 −5 7 6 −5 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 −2 1 7 6 −2 1 7 6 −2 1 7 6 −2 1 7 4 5 4 5 4 5 4 5 −2 −2 −2 −2 3 Prior Work • Arnol’d, V.I.: On matrices depending on parameters. Russian Mathematical Surveys 26(2) (1971) 29–43 • Chen, G.: Computing the normal forms of matrices depending on parameters. In: Proceedings of ISSAC 1989, ACM (1989) 242–249 4 Prior Work • Ballarin, C., Kauers, M.: Solving parametric linear systems. ACM SIGSAM Bulletin 38(2) (2004) 33–46 • Broadbery, P.A., Gómez-Díaz, T., Watt, S.M.: On the implementation of dynamic evaluation. In: Proceedings of ISSAC 1995, ACM (1995) 77–84 • Diaz-Toca, G.M., Gonzalez-Vega, L., Lombardi, H.: Generalizing Cramer’s rule. SIAM Journal on Matrix Analysis and Applications. 27(3) (2005) 621–63 • Kapur, D.: An approach for solving systems of parametric polynomial equations. Principles and Practices of Constraint Programming (1995) 217–244 • Sit, W.Y.: An algorithm for solving parametric linear systems. Journal of Symbolic Computation. 13(4) (1992) 353–394 5 Regular Chain Theory Notation • Let K be a field, • Let X1 < ··· < Xs be s ≥ 1 ordered variables, • K[X1;:::; Xs] = K[X] is the ring of polynomials in X = X1;:::; Xs • For a non-constant p 2 K[X], mvar(p) denotes the greatest variable in p • The leading coefficient of p w.r.t. mvar(p) is called the initial denoted init(p) 6 Regular Chain Triangular Set A set T of non-constant polynomials in K[X] is called a triangular set if for all p; q 2 T with p =6 q, mvar(p) =6 mvar(q) : Saturated Ideal h i 1 The saturated ideal, sat(T) of T is the ideal T : hT , where hT is the product of the initials of the polynomials in T Regular Chain T is a regular chain if T = ? or T = T0 [ ftg for t 2 T with mvar(t) maximum such that • T0 is a regular chain • init(t) is regular modulo sat(T0) 7 Regular Chain Solution Set s For a set of polynomials F ⊂ K[X1;:::; Xs], V(F) ⊆ K consists of all common roots of the polynomials in F Quasi-component The quasi-component of a triangular set is defined as W(T) = V(T) n V(hT) where hT is the product of the initials of T. 8 Regular System Regular System Given a regular chain T ⊂ K[X] and a set of polynomials H ⊂ K[X], the pair [T; H] is a regular system if each polynomial h 2 H is regular modulo sat(T). The zero set of [T; H] is denoted Z(T; H) = W(T) n V(H). 9 Square-free Factorization over a Regular System The function1 Squarefree_RC(p; T; H) computes a set of triples (( ;:::; ); ; ) ≤ ≤ [ ; ];:::; [ ; ] bi;1 bi;`i Ti Hi with 1 i e such that T1 H1 Te He are regular systems forming a triangular decomposition of Z(T; H). For all 1 ≤ i ≤ e: 1. b ;:::; b are polynomials with the same main variable v = mvar(p) such i;1 Qi;`i ≡ `i j ( ) that p j=1 bi;j mod sat Ti , 2. All discriminants disc(bi;j; v) are regular modulo sat(Ti) 3. All resultants res(b ; b ; v) are regular modulo sat(T ) Q i;j i;k i `i j 4. j=1 bi;j is a square-free factorization of p modulo sat(Ti) 1Chen, C., Moreno Maza, M.: Algorithms for computing triangular decomposition of polynomial systems. Journal of Symbolic Computation, 47(6) (2012) 610–642 10 Linear Algebra Theory Companion Matrix n n−1 Let (x) = x + an−1x + ··· + a1x + a0. The Frobenius companion matrix of (x) is 2 3 0 0 ··· 0 −a 6 0 7 6 ··· − 7 61 0 0 a1 7 6 ··· − 7 C( (x)) = 60 1 0 a2 7 6 . 7 4 . .. 5 0 0 ··· 1 −an−1 11 Frobenius Form A matrix F 2 Kn×n is in Frobenius form if 2 3 C( (x)) 6 1 7 6 7 6 C( 2(x)) 7 F = 6 . 7 4 .. 5 C( m(x)) j − and i+1 i for i = 1;:::; m 1. 12 Properties of Frobenius Form • The polynomials i are called the invariant factors of F. • is the minimal polynomial of F Q1 m • i=1 i is the characteristic polynomial of F • There exists a non-singular matrix Q such that F = Q−1AQ is in Frobenius form • The geometric multiplicity of a eigenvalue λ of F is the number of invariant factors it is a solution of. 13 Computing the Frobenius Form • Ozello, P. (1987). Calcul exact des formes de Jordan et de Frobenius d’une matrice (Doctoral dissertation, Université Joseph-Fourier-Grenoble I). • Storjohann, A. An O(n3) algorithm for the Frobenius normal form. In Proceedings of ISSAC 1998, ACM (1998) 101–105. 14 Jordan Block Matrix A Jordan block is a matrix of the form 2 3 λ 1 6 7 6 λ 7 6 1 7 6 . 7 JBMm(λ) = 6 .. .. 7 6 7 4 λ 15 λ • JBMm(λ) has a single eigenvalue λ • λ has geometric multiplicity 1 15 Jordan Canonical Form A matrix in Jordan canonical form is of the form 2 3 JBMm1 (λ1) 6 . 7 4 .. 5 JBMm` (λ`) 16 Frobenius Form to Jordan Form 2 3 C( (x)) 6 1 7 6 7 6 C( 2(x)) 7 F = 6 . 7 4 .. 5 C( m(x)) The Jordan canonical form of F is 2 3 JCF(C( (x))) 6 1 7 6 7 6 JCF(C( 2(x))) 7 6 . 7 4 .. 5 JCF(C( m(x))) 17 Jordan Form of a Companion Matrix The JCF of a companion matrix C( (x)) is 2 3 JBM (λ ) 6 m1 1 7 6 (λ ) 7 6 JBMm2 2 7 JCF(C( (x))) = 6 . 7 4 .. 5 JBMm` (λ`) where Y` − mi (x) = (x λi) i=1 6 6 with λi = λj for i = j. 18 JCF Over a Splitting Field Example Consider the companion matrix 2 3 0 0 −1 6 7 A = 41 0 3 5 0 1 1 Maple computes the JCF as 2 p p 3 3 −1=3 26 + 6 33 + 8=3 p 1 p + 1=3 0 0 6 3 7 6 26+6 33 ( ) 7 6 p p p p p 7 6 3 3 7 − p 1 p − − − p 1 p JCF(A) = 6 0 1=6 26 + 6 33 4=3 3 + 1=3 i=2 3 1=3 26 + 6 33 8=3 3 0 7 6 26+6 33 26+6 33 7 6 ( ) 7 4 p p p p p 5 3 3 − p 1 p − − p 1 p 0 0 1=6 26 + 6 33 4=3 3 + 1=3 + i=2 3 1=3 26 + 6 33 8=3 3 26+6 33 26+6 33 19 Example 2 3 1 − y − y 0 0 6 2 1 7 JCF(A) = 4 0 y1 0 5 0 0 y2 where (y1; y2) is any point in the zero set V(T) of 8 < 2 2 y + (y2 − 1)y1 + y − y2 + 3 T = 1 2 : 3 − 2 y2 y2 + 3y2 + 1 20 Objective Given a monic polynomial (x) 2 K[x], and its companion matrix C( (x)) 2 Kn×n, find a regular chain T ⊂ K[y1;:::; ys] such that the JCF of C( (x)) can be computed over the splitting field2 generated by T. i.e. any point p 2 V(T) defines the splitting field of T: L = K(p). 2Landau, S. (1985). Factoring polynomials over algebraic number fields. SIAM Journal on Computing, 14(1), 184-195. 21 Algorithm Input: A square-free polynomial p 2 K[x] Algorithm: L K f g P f g F f g 1. Initialize i := 1, yi := x, := , T := , = and := p . 2. While F is not empty do 2 F L a. Pick a polynomial f(yi) over , b. Let αi be a root of f(yi) L L c. Replace by (αi) [ f g d.
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