On Primary Decomposition and Polynomial of a Matrix

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 The goal of this paper is to study some unknown questions on the primary decomposition of matrices over a field K and to give the analogous of some well known results of spectral, algebraic and geometric multiplicity order of an eigenvalue to any P-component of the characteristic polynomial C A of a matrix A over a field K. More precisely we compute the dimension of the kernel of a polynomial of a square matrix A over any arbitrary commutative field K in terms of its invariant factors. As an application we determine the value of the P-algebraic and P-geometric multiplicity order of any P-component of the characteristic polynomial C A of a matrix A.


Introduction
Let K be a field.Let A ∈ M n (K) and P be an irreducible polynomial of K[X].We will say that A is P -primary matrix if the characteristic polynomial C A of A is a power of P .The Primary decomposition Theorem states that if A ∈ M n (K) is a non zero matrix and m A (X) = s i=1 P α i i is the prime decomposition of its minimal polynomial m A (X) then the matrix A is similar to a block diagonal of P -primary matrices diag(A 1 , A 2 , ..., A s ).The dimension of sequence vector spaces Ker P s (A)-is unknown.
In the first part of this paper, we use some deep results on module theory over a PID to compute the dimension of the kernel of a polynomial of a square matrix A over a commutative field K in terms of its invariant factors.
In the second part, we give the analogous of some well known results of spectral, algebraic and geometric multiplicity order of an eigenvalue, to any Pcomponent of the characteristic polynomial C A of a matrix A over any arbitrary commutative field K. Some new results on the P -algebraic and P -geometric multiplicity order are also established.

Preliminary Notes
Let K be a field.Let M be a finite dimension vector space over K and f a K-endomorphism of M. The vector space M is endowed by a structure of K[X]-module via the endomorphism f by X.m = f (m) for any m ∈ M. We will denote by M f the K[X]-module on M induced by f .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 [ [6], §2, p. 556],[ [8], § 14], [ [1], p. 235] and [3] ): Theorem 2.1 (Rational canonical form) Let M be a finite-dimensional vector space over a field K and f be a K-endomorphism of M. Let M f be the K[X]-module induced by f then there exists a unique sequence of polynomials q 1 , • • • , q r such that: and • q r = m f (X) the minimal polynomial of f and r i=1 q i = c f (X) the characteristic polynomial of f .The ascending sequence of polynomials q 1 , • • • , q r are unique and called the invariant factors of f .If q 1 , • • • , q r are the invariant factors of f then we will write Let A ∈ M n (K) be a no zero matrix, and for any linear transformation that has matrix A relative to some basis, we denote M A the K[X]-module induced by A. Then by theorem2.1: Let K be a field.Let A ∈ M n (K) and P be an irreducible polynomial of K[X].We will say that A is P -primary matrix if the characteristic polynomial C A of A is a power of P .

Proposition 2.2 (Primary decomposition Theorem
Then the subspaces E i are invariant under A and A is similar to a block diagonal of P -primary matrices diag(A 1 , A 2 , ..., A s ).

Proof. See [[7], Theorem 1.5.1,p29].
Throughout this paper, E is a finite-dimensional vector space over a field K.If f ∈ End K (E), m f and C f stand respectively for the minimal and the characteristic polynomial of f .

Main Results
This is the main result of this paper.

Theorem 3.1 Let K be a field. Let A ∈ M n (K) be a non zero matrix and
for any P ∈ K[X].In particular dim K KerA is the number of i such that q i (0) = 0.
To prove this Theorem we need the following lemmas Lemma 3.2 Let u be an endomorphism of a finite dimensional vector space Proof.Easy to prove (see [[8], Proposition 1. 3. 2] and [5]).

Lemma 3.4 Let A ∈ M n (K) and let M A be the K[X]-module induced by A. If M A K[X]/(q). Let P ∈ K[X],then Ker(P (A)) Ker P (X)
where P (X) : K[X]/(q) → K[X]/(q),T → P (X).T

Lemma 3.5 Let A ∈ M n (K) and let M A be the K[X]-module induced by
By lemma 3.4 and lemma 3.3 we have Ker P (X) K[X]/(D) where D = gcd(P, q).Now let's give the proof of the theorem 3.1 Proof.Let E be a K-vector space of finite dimension.Let f ∈ End K (E) and B a basis of E such that mat B (f ) = A. The space E can be viewed as a K -modules, where q 1 , q 2 , ..., q r are the invariant factors of A .Hence E = ⊕ r i=1 E i where E i 's are f -invariant subspaces and where f i = res E i f .So it turns to study the case where f admits one invariant factor (A is companion).By lemma 3.5 KerP

Generalized algebraic and geometric multiplicity order
Let K be a field.Let Q be a polynomial of K[X] and P be an irreducible polynomial of K[X] which occur in the prime decomposition of Q.We will say that the power polynomial P s is the coprime with P .The integer s is said the P -valuation of Q and will be denoted by υ P (Q).
In order to give the analogous of some well known results of spectral, algebraic and geometric multiplicity order of an eigenvalue.We introduce the P -algebraic and P -geometric multiplicity order relative to any P -component of the characteristic polynomial C A of the matrix A. • The P -algebraic multiplicity order of the matrix A (or the algebraic multiplicity order of A at the factor P ) is dim K KerP (A) υ P (C A ) .
• The P -geometric multiplicity order of the matrix A (or the geometric multiplicity order of A at the factor P ) is dim K KerP (A).
Throughout this work we will follow the notations used by the authors of [1]: 1) ν alg (P ) denote the P -algebraic multiplicity order of the matrix A.
2) ν geom (P ) denote the P -geometric multiplicity order of the matrix A.
where k is the number of i such that s i ≤ l.
Proof.Indeed, by theorem 3.1 dim K KerP l (f ) = r i=1 deg (gcd(P l , q i )) = r i=1 inf (l, s i )degP so we deduce the result.Proof.Indeed, if t = υ P (C f ) and IF (f ) = (q 1 , • • • , q r ) are the invariant factors of f and l ≥ s = s r then l ≥ s i for all i = 1, • • • , r so by proposition 4.2 r=k hence dim K KerP l (f ) = ( r i=1 s i )degP = tdegP since r i=1 s i = t.

Corollary 4.4
Let A ∈ M n (K).Let C A be the characteristic polynomial of A. If P is an irreducible monic factor of C A then P -algebraic multiplicity order of the matrix A is υ P (C A )degP .
Proof.Indeed, let f be the endomorphism canonically associated to A. By the corollary 4.3 and since υ P (C f ) ≥ υ P (m f ) we have dim K KerP t (f ) = tdegP where t = υ P (C f ).
Let f ∈ End K (E) and N k = Kerf k .As E is a finite dimension vector space over K, the sequence N k is stationary.It is well known that if N k = N k+1

Definition 4 . 1
Let A ∈ M n (K).Let C A be the characteristic polynomial of the matrix A. If P is an irreducible monic factor of C A then

Corollary 4 . 3
Let f ∈ End K (E) and P ∈ K[X] be an irreducible monic factor of C f .Let s = υ P (m f ).Then dim K KerP l (f ) = υ P (C f ) degPfor any positive integer l ≥ s.