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 ∈ 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 ∈ 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 ∈ Mn(K) and B ∈ Mn(K) two square matrices.

1. A and B are similar if there exist a square invertible matrix C ∈ 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   A1 0 ··· 0    0 A2 ··· 0   . . . .   . . .. .    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 ∈ 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], §2, 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 | 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 ∈ 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 | 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 ∈ E, k X u(x) = ui(xi) i=1 Pk where x = i=1 xi (xi ∈ 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 ϕ : i=1 Mi → i=1(Mi)ui ,(xi)i 7→ i=1 xi. ϕ is a K[X]-isomorphism. ϕ(X.(xi)i) = ϕ((X.xi)i) = ϕ((u(xi))i) = ϕ((ui(xi))i) = Pk Pk Pk i=1 ui(xi) = u( i=1 xi) = X. i=1 xi = Xϕ((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 ∈ 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 ∈ R be a prime. Define the p-component M(p) of M by

n + M(p) = {x ∈ M : Ann(x) =< p > for some n ∈ Z }.

If M = M(p) then M is said to be p-primary and M is primary if it is p-primary for some p ∈ 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 = p∈P M(p).

Proposition 3.3 Let M be a torsion module over a PID ring R and let L L p ∈ R be a prime. If M = i∈I Mi then M(p) = i∈I Mi(p).

Proof. Easy.

Proposition 3.4 Let K be a field. Let A ∈ 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) = gcd{a, b}, then

{c ∈ R/bR | ac = 0}' 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 ∈ R we have M(p) = {x ∈ M | pυp(a)x = 0}. 864 S. Bouarga and M. E. Charkani

υp(a) Proof. Set Mpυp(a) := {x ∈ M | p x = 0}. The inclusion Mpυp(a) ⊂ M(p) is obvious. Let x ∈ M(p) so there exists t ∈ N such that ptx = 0. We have a = pυp(a)q with gcd(p, q) = 1. So gcd(pt, q) = 1. So there ex- ist u, v ∈ R such that upt + vq = 1. So x = uptx + vqx = vqx. So

υp(a) υp(a) υp(a) p x = p vqx = p qxv = axv = 0. Hence x ∈ Mpυp(a) .

Corollary 3.8 Let R be a PID and M an R-module. If M = R/aR then for any prime p ∈ R we have M(p) ' R/pυp(a)R.

Proof. Indeed if M = R/aR then annRM = aR. So M(p) ' Mpυp(a) and by lemma 3.6 if M = R/aR then M(p) ' R/dR where d = pgcd(a, pυp(a)) = pυp(a).

3.2 Primary decomposition and invariant factors

Let K be a field. Let A ∈ 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 .

A P - primary matrix A is said of type λ = (λ1, ··· , λr) if its invariant factors are IF (A) = (P λ1 , ··· ,P λr ). The next theorem is an easy consequence of structure theorems for K[X]- modules. For proof of these see [[3], A.VII.31], [6].

Theorem 3.9 (Primary Decomposition) Every matrix A ∈ Mn(K) is sim- ilar to a matrix of the form ⊕P AP . where AP is a P-primary matrix, and the sum is over the irreducible factors of the characteristic polynomial of A. More- over, for every P, the similarity class of AP is uniquely determined by the similarity class of A.

Definition 3.10 Let u be an endomorphism of a finite dimensional vector Qr ti space over K, and Cu(X) = i=1 Pi the primary decomposition of its charac- ti teristic polynomial. Let Ei = KerPi (u), ui = resEi u is called the Pi−primary component of u.

Let A ∈ Mn(K) and u a linear transformation such that matB(u) = A relatively to some basis B. If ui is the Pi−primary component of u. Then On invariant factors of primary components 865

Ai = matB(ui) is called the Pi−primary component of A relatively to the basis B.

Let A ∈ Mn(K) be a non zero matrix. If IF (A) = (q1, ··· , qr) define ( q if k ≤ r IF (A) := r−k+1 k 1 if k > r

In this case IF (A) = (IFr(A), ··· ,IF1(A)).

Proposition 3.11 Let K be a field. Let A ∈ Mn(K) be a non zero matrix. If A Se diag(A1, ··· ,Al) such that Ai’ s are Pi-primary matrices for i = 1, ··· , l, and r = max(ri)1≤i≤l where ri is the number of invariant factors of Ai. Then the invariant factors of A are IF (A) = (IFr(A), ··· ,IF1(A)) where IFk(A) = Ql i=1 IFk(Ai) for 1 ≤ k ≤ r.

Ll Proof. If A Se diag(A1, ··· ,Al) then by lemma 2.5 MA ' i=1 MAi and Lri by theorem 2.1 MAi ' j=1 K[X]/(IFj(Ai)).   Ll Lri Lr Ll Hence MA ' i=1 j=1 K[X]/(IFj(Ai)). Hence MA ' j=1 i=1 K[X]/(IFj(Ai)) . Lr Ql By using the Chinese Remainder Theorem we have MA ' j=1 K[X]/( i=1 IFj(Ai)) ' Lr Ql Ql j=1 K[X]/(IFj(A)). We have IFj+1(A) = i=1 IFj(Ai) = i=1 qri−j(Ai) Ql Ql and IFj(A) = i=1 IFj(Ai) = i=1 qri−j+1(Ai) and since qri−j(Ai) | qri−j+1(Ai) for 1 ≤ i ≤ l then IFj+1(A) divide IFj(A) for 1 ≤ j ≤ r − 1. Hence

IF (A) = (IFr(A), ··· ,IF1(A)).

Proposition 3.12 Let K be a field. Let A ∈ Mn(K) be a non zero matrix.

Let P be an irreducible monic factor of CA. Let B be the P -primary component υP (qi) matrix of A. Then IF (B) = (P )1≤i≤s where s = max{i | P divides qi}.

In other words B is a P -primary matrix of type (υP (q1), ··· , υP (qs))

Proof. Let MA be the K[X]-module induced by A. Let IF (A) = Lr (q1, ··· , qr). So MA ' i=1 K[X]/(qi) with qi | qi+1 for any i ≤ r − 1. Hence Lr MA(P ) ' i=1(K[X]/(qi))(P ). By corollary 3.8 we have (K[X]/(qi))(P ) = (P υP (qi)) for any i ≤ r − 1.

Lr Ls υP (qi) Hence MA(P ) ' i=1(K[X]/(qi))(P ) ' i=1(K[X]/(P ). As qi | qi+1 υP (qi) υP (qi+1) υP (qi) then υP (qi) ≤ υP (qi+1). So P | P . So IF (B) = (P )1≤i≤s.

Recall that if A ∈ Mn(K) is a non zero matrix and IF (A) = (q1, ··· , qr) its invariant factors then the sequence of non negative integers di = deg qi is a partition of n called the partition associated to invariant factors of A. 866 S. Bouarga and M. E. Charkani

α When A is a P - primary matrix then mA(X) = P where P is an irreducible β polynomial of K[X] and CA(X) = P with α ≤ β. Further there exists λi a partition λ = (λ1, ··· , λr) of β such that λr = α and qi = P for any i. We will say that the partition λ = (λ1, ··· , λr) is the exponent partition associated to the P - primary matrix A. A nilpotent matrix N is said of type

λ1 λr λ = (λ1, ··· , λr) if its invariant factors are IF (N) = (X , ··· ,X ). As a nilpotent matrix N is a X- primary matrix then a nilpotent matrix N is of type λ = (λ1, ··· , λr) if λ = (λ1, ··· , λr) is the exponent partition associated to the matrix N.

Proposition 3.13 Let P be an irreducible polynomial of K[X] and A ∈

Mn(K) be a P - primary matrix. Let λ = (λ1, ··· , λr) be the exponent partition associated to the P - primary matrix A. Let B = P (A) be the polynomial matrix of A. Then B is a nilpotent matrix of type λ = (λ1, ··· , λ1, ··· , λr, ··· , λr). In | {z } | {z } degP degP other words the invariant factors of B are IF (B) = (Xλ1 , ··· ,Xλ1 , ··· ,Xλr , ··· ,Xλr ). | {z } | {z } degP degP

Proof. Notice that if P s(A) = 0 for some nonzero positif integer s then s B = 0. If λ = (λ1, ··· , λr) is the exponent partition associated to the P - primary matrix A. Then Bλr = 0 and Bλr−1 6= 0. So B is a nilpotent matrix of index λr. We conclude the result by using Theorem 3.1 and Proposition 4.2 from [2].

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Received: September 15, 2014, Published: November 24, 2014