International Journal of Algebra, Vol. 8, 2014, no. 18, 859 - 867 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2014.4991 On Invariant Factors of Primary Components of Square Matrices S. Bouarga Department of Mathematics Faculty of Science and Technology, FST Fez saiss Fez, Morocco M. E. Charkani Department of Mathematics Faculty of Sciences, Dhar-Mahraz P.O. Box 1796, Atlas-Fez, Morocco Copyright c 2014 S. Bouarga and M. E. Charkani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we introduce the P-primary component of a square matrix A over any arbitrary commutative field K, where P is an ir- reducible polynomial of K[X]. We use some deep results on module theory over a PID to establish the links between the invariant factors of A and the invariant factors of its primary component. We also prove that if A a P-primary matrix. Then B=P(A) is a nilpotent matrix of type λ = (λ1; ··· ; λ1; ··· ; λr; ··· ; λr). Where λ = (λ1; ··· ; λr) is the | {z } | {z } degP degP exponent partition associated to the invariant of A. Keywords: Primary component of a square matrix, invariant factors, ex- ponent partition 860 S. Bouarga and M. E. Charkani 1 Introduction Let K be a field. Let A 2 Mn(K) and P be an irreducible polynomial of K[X]. We will say that A is P - primary matrix if the characteristic polynomial CA of A is a power of P . The Primary decomposition Theorem states that if A 2 Mn(K) is a non zero matrix then A is similar to a block diagonal of P - primary matrices diag(A1;A2; :::; As). The aim of this paper is to investigate the relation between the invariant factors of Ai and the invariant factors of A. Notations: 1.1 Let K be a field. Let A 2 Mn(K) and B 2 Mn(K) two square matrices. 1. A and B are similar if there exist a square invertible matrix C 2 GLn(K) such that B = CAC−1. We will write A SBe Lm 2. If Ai' s for 1 ≤ i ≤ m are square matrices, then i=1 Ai = diag(A1; ··· ;Am) denotes the block diagonal matrix 0 1 A1 0 ··· 0 B C B 0 A2 ··· 0 C B . C B . .. C @ A 0 0 ··· Am ith 3. If A is a non zero matrix then qk(A) or qk denotes the k invariant factor of A for all integer k ≥ 1. 2 Preliminary Notes 2.1 K[X]-module induced by an endomorphism Let K be a field. Let M be a vector space over K and u a K-endomorphism of M. The vector space M is endowed by a structure of K[X]-module via the endomorphism u by X:m = u(m) for any m 2 M. We will denote by Mu the K[X]-module on M induced by u. As the ring K[X] is a PID, then by applying the structure theorem of finitely generated torsion modules over a PID, the very useful following theorem is deduced (see [[5], x2, p. 556] and [[1], p. 235]): On invariant factors of primary components 861 Theorem 2.1 (Rational canonical form) Let M be a finite-dimensional vector space over a field K and u be a K-endomorphism of E. Let Mu be the K[X]-module induced by u then there exists a unique sequence of polynomials q1; ··· ; qr such that: K[X] K[X] K[X] Mu ' ⊕ ⊕ · · · ⊕ (q1) (q2) (qr) and • qi j qi+1, for i ≤ r − 1 Qr • qr = mu(X) the minimal polynomial of u and i=1 qi = cu(X) the char- acteristic polynomial of u. The ascending sequence of polynomials q1; ··· ; qr are unique and called the invariant factors of u. If q1; ··· ; qr are the invariant factors of u then we will write IF (u) = (q1; ··· ; qr). Let A 2 Mn(K) be a no zero matrix, and for any linear transformation that has matrix A relative to some basis, we denote MA the K[X]-module induced by A. Then by theorem 2.1: K[X] K[X] K[X] MA ' ⊕ ⊕ · · · ⊕ (q1) (q2) (qr) Qr such that qi j qi+1, qr = mA(X) the minimal polynomial of A and i=1 qi = cA(X) the characteristic polynomial of A. The sequence of polynomials q1; ··· ; qr are called the invariant factors of A and we write IF (A) = (q1; ··· ; qr). The in- variant factors of A are unique up similarity. Indeed if q1; ··· ; qr are the invari- ant factors of A then A is similar to a block diagonal matrix diag(A1;A2; :::; Am) where Ai = Comp(qi) is the companion matrix of qi. Remark 2.2 Let r be the number of invariant factors of A, mA be the minimal polynomial and CA the characteristic polynomial of A. Then q1 = r ··· = qr if and only if CA = (mA) . 862 S. Bouarga and M. E. Charkani 2.2 Direct Sums and Matrices Let u be an endomorphism of a finite dimensional vector space E over K. If k E = ⊕i=1Ei, where each Ei is u−invariant. Then we may write, for every x 2 E, k X u(x) = ui(xi) i=1 Pk where x = i=1 xi (xi 2 Ei; i = 1; ··· ; k) and ui = resEi u the restriction of u to Ei. In this case u is called direct sum of ui = resEi u (i = 1; ··· ; k) and k is written u = ⊕i=1ui = u1 ⊕ · · · ⊕ uk (see [[4],p 144]). Lemma 2.3 Let u be an endomorphism of a finite dimensional vector space k Lk M over K. If u = ⊕i=1ui with ui = resMi u and M = i=1 Mi, where each Mi is u−invariant subspace. Then k M Mu ' (Mi)ui i=1 Lk Lk Pk Proof. Indeed let ' : i=1 Mi ! i=1(Mi)ui ,(xi)i 7! i=1 xi. ' is a K[X]-isomorphism. '(X:(xi)i) = '((X:xi)i) = '((u(xi))i) = '((ui(xi))i) = Pk Pk Pk i=1 ui(xi) = u( i=1 xi) = X: i=1 xi = X'((xi)i) Proposition 2.4 Let u be an endomorphism of a finite dimensional vec- tor space M over K. Let A be the matrix of u in a basis B of M. Then A Se diag(A1; ··· ;Ak) if and only if u can be decomposed into a direct sum of k endomorphisms. Proof. See [[4],Theorem 1,p 146] Corollary 2.5 Let K be a field. Let A 2 Mn(K) be a non zero matrix. If Lk A Se diag(A1; ··· ;Ak) then MA ' i=1 MAi . Proof. Let u be an endomorphism of a finite dimensional vector space M over K, such that A is the matrix of u in a basis B of M. We have k A Se diag(A1; ··· ;Ak). So by proposition 2.4 u = ⊕i=1ui and matui = Ai. Lk Lk Then by lemma 2.3 Mu ' i=1(Mi)ui . Hence MA ' i=1 MAi . On invariant factors of primary components 863 3 Primary decomposition 3.1 Primary component of module Definition 3.1 ([1], p. 162) Let M be a torsion module over a PID ring R and let p 2 R be a prime. Define the p-component M(p) of M by n + M(p) = fx 2 M : Ann(x) =< p > for some n 2 Z g: If M = M(p) then M is said to be p-primary and M is primary if it is p-primary for some p 2 R. Proposition 3.2 ( [5], [1]) Let M be a torsion module over a PID ring R L and let P the set of primes in R. Then M = p2P M(p). Proposition 3.3 Let M be a torsion module over a PID ring R and let L L p 2 R be a prime. If M = i2I Mi then M(p) = i2I Mi(p). Proof. Easy. Proposition 3.4 Let K be a field. Let A 2 Mn(K) be a non zero matrix. Let P be an irreducible polynomial. If A Se diag(A1; ··· ;Ak) then MA(P ) ' Lk i=1 MAi (P ) where M(P ) is the P -component of the R-module M. Proof. Indeed, if A Se diag(A1; ··· ;Ak) then by corollary 2.5 MA ' Lk Lk i=1 MAi . Hence by proposition 3.3 MA(P ) ' i=1 MAi (P ). Notation: 3.5 If pska (the highest power of p dividing a) we write s = υp(a). Lemma 3.6 Let R be a PID and let a; b be nonzero elements of R. If d = (a; b) = gcdfa; bg, then fc 2 R=bR j ac = 0g ' R=dR: Proof. See [[2],Lemma 3.3] Lemma 3.7 Let R be a PID and M an R-module. If AnnRM = aR then for any prime p 2 R we have M(p) = fx 2 M j pυp(a)x = 0g. 864 S. Bouarga and M. E. Charkani υp(a) Proof. Set Mpυp(a) := fx 2 M j p x = 0g. The inclusion Mpυp(a) ⊂ M(p) is obvious. Let x 2 M(p) so there exists t 2 N such that ptx = 0. We have a = pυp(a)q with gcd(p; q) = 1. So gcd(pt; q) = 1.