P-Canonical Forms and Drazin Inverses 11

-CANONICAL FORMS AND DRAZIN INVERSES P M. MOUÇOUF k Abstract. In this paper, P-canonical forms of (A )k (or simply of the matrix A) are defined and some of their properties are proved. It is also shown how we can deduce from them many interesting informations about the matrix A. In addition, it is proved that the P-canonical forms of A can be written as a sum of two parts, the geometric and the non-geometric parts of A, and that the P-canonical form of the Drazin inverse Ad of A can be deduced by simply plugging −k for k in the geometric part of A. Finally, several examples are provided to illustrate the obtained results. Department of Mathematics, Faculty of Science, Chouaib Doukkali University, Morocco Email: [email protected] 1. Introduction Let A be a nonsingular matrix over a field F . It is a well known fact that, for many numerical examples of matrices A, replacing k with k k − 1 in certain forms of A , one can obtain the kth power of the inverse A− of A. However, this fact is not proven in general. The problem here is that the kth power of a matrix can be represented in several forms. In fact, It can happen that in certain forms of Ak it is not even possible to substitute k 1 and, in other forms we can substitute k 1 but = − 1 = − we does not obtain A− . For example, Let A be an r-circulant matrix. The expressions of Ak given in Theorem 4.1 of [16] and Theorem 3.1 arXiv:2007.10199v5 [math.RA] 27 Sep 2021 of [17] do not provide A 1 when we plug into them k 1. − = − In this paper, we consider an arbitrary matrix A, singular or nonsingular, with entries in a field F and we prove that if we plug k for k into − the geometric part of Ak we get the kth power of the Drazin inverse ( )k of A. In order to avoid any confusion that may arise by using this plugging-in operation, we begin by showing that the representations of Ak into them we plug k for k, are canonical. ( )k − An element a of an associative ring is said to have a Drazin inverse [8] 2020 Mathematics Subject Classification. 15A09, 15AXX, 05A10, 11B37. Key words and phrases. Powers of matrices, P-Canonical forms, Linear recurrence sequences, Binomial coefficients, Drazin inverses. 1 2 M. MOUÇOUF if there exists an element b, written b a , such that = d akba ak, ab ba, bab b = = = for some nonnegative integer k. It is well known that any element of any associative ring has at most one Drazin inverse (see [8]). The theory of Drazin inverse has been extensively studied and suc- cessfully applied in many fields of science such as functional analysis, matrix computations, combinatorial problems, numerical analysis, sta- tistics, population models, differential equations, Markov chains, con- trol theory, and cryptography, etc. [1,4,5,13,14,25]. For this rea- son, many interesting properties of the Drazin inverse have been obtained [3, 7, 8, 14, 22] and a variety of direct and iterative methods have been developed for computation of this type of generalized inverse [2, 7, 10, 12, 20–25]. It should be noted that the method presented herein provides a closed- form formula for the kth power of the Drazin inverse Ad of A and gives other interesting information on the matrix A such as the min- imal polynomial and the Jordan-Chevalley decomposition of A. But for this, a closed-form formula for the kth power of the matrix A is required and this can be done by any of the well-known methods such as Kwapisz’s method [11]. The organization of this paper is as follows. In Section 2, some alge- braic results are established for the F -algebra of all linear recurrence sequences over F whose characteristic polynomials split over F , and then, the definitions of the geometric part and the non-geometric part of a linear recurrence sequence are given. Results obtained there will be used in Section 3 to derive similar results for the set of all sequences of matrices, with coefficients in F , satisfying linear homogeneous recurrence relations with constant coefficients in F . In addition, some interesting properties are shown about the -canonical forms of matri- P ces defined in this section. Lastly, section 4 shows that the -canonical P forms of the Drazin inverse of a matrix A can be deduced from those of the matrix A by a simple plugging-in operation. 1.1. Notations. Throughout the paper, we use the following notations: F is a field and F is the set of all sequences s sk k 0 over F . ● C = ( ) ⩾ It is well known that is an F -algebra under componentwise CF addition, multiplication and scalar multiplication. Sequences in this paper are written in bold symbol. ● k Λi, i 0, is the element i k 0 of F . ● ⩾ ( k) ≥ C Λi, i 0, is the element − k 0 of F . ● ̃ ⩾ ( i ) ≥ C P-CANONICALFORMS ANDDRAZININVERSES 3 is the set Λ i 0 . ● T { i~ ⩾ } Γ is the element 0, 1, 2,... of . ● ( ) CF is the set Γi i 0 . ● H { ~ ⩾ } If is a subring of a ring and L is subset of , then L ● R D D R⟨ ⟩ denotes the submodule spanned by L. If is a subalgebra of an algebra and L is a subset of , ● R K K then L denotes the subalgebra of obtained by adjoining to K the set[ ]L. K Mq G is the set of q q matrices over an F -algebra G. ● ( ) k × ∗ λ λ k 0 λ F,λ 0 the set of all nonzero geometric ● S = = ⩾ ∈ ≠ sequences.{ ( ) ~ } 0 i N,0 k δ . ● S○ = i ∈CF ∈ i = ik { . ~ ( ) } ● S=S∗ ∪S○ F denotes the F -vector spaces spanned by . It is well known ● S S that F is the set of all linear recurrence sequences over F whose characteristicS polynomials are of the form XmP X where the polynomials P X are square-free and split over F( with) nonzero constant terms.( ) F ∗ denotes the F -vector spaces spanned by ∗. It is well known ● S S that F ∗ is the set of all linear recurrence sequences over F whoseS characteristic polynomials are square-free and split over F with nonzero constant terms. F ○ denotes the F -vector spaces spanned by ○. It is well known ● S S that F ○ is the set of all linear recurrence sequences over F whose characteristicS polynomials split over F and have zero as their only root. q F denotes the subalgebra Mq F ∗ of Mq F . ● G ( ) ( S )[T ] (C ) q F denotes the subalgebra Mq F of Mq F . ● L ( ) ( S )[T ] (C ) It is well known from the general theory of linear recurrence sequences that , ∗ and ○ are, respectively, bases of F , F ∗ and F ○ . ● S S S S S S F , F ∗ and F ○ are subalgebras of F . More precisely, the set ● S S S C ∗ is a group, and hence F ∗ is exactly the group algebra of ∗ S S S over F . F ○ and F ∗ are supplementary vector spaces relative to F , ● S S S i.e., F F ○ F ∗ . S = S ⊕ S 2. Canonical forms for linear recurrence sequences k Let Λi i 0 be the set of all sequences Λi k 0 and T = ⩾ = i ≥ consider the{F -vector~ } space F spanned by . Then( we have) the ⟨T ⟩ T following result 4 M. MOUÇOUF Proposition 2.1. Let F be a field. Let ̃ be the set of all sequences k T Λi − k 0 and consider the sequence Γ 0, 1, 2,... Then we have ̃ = ( i ) ≥ = ( ) 1. F is a subalgebra of and is a basis of F . ⟨T ⟩ CF T ⟨T ⟩ 2. is a basis of F , and the linear automorphism χ of F which T̃ ⟨T ⟩ ⟨T ⟩ maps Λ to Λ is an involution of the algebra F . i ̃i ⟨T ⟩ 3. F and F ∗ are F -linearly disjoint. ⟨T ⟩ S 4. If F is of characteristic 0, then the sequence Γ is transcendental over the ring F ∗ . S 5. F F ○ F ∗ . S [T ] = S ⊕ S [T ] Proof. 1. Using the following formula (due to Riordan [19]) i j k k + m i k ij = mQi i m jm = − i i j m k + − = mQ0 m, i m, j mi j m = − − + − we can easily deduce that ΛiΛj is a linear combination of elements of for all i, j. Then F is a subalgebra of . More precisely, F T ⟨T ⟩ CF ⟨T ⟩ is the set of all linear recurrence sequences over F with charateristic polynomials split over F and have one as their only root. It is well known that is a basis of F . T ⟨T ⟩ 2. Let i N. It is clear that Λ̃0 Λ0 and then Λ̃0 F . Suppose that ∈ = i k i 1 ∈ ⟨T ⟩ i 1. Then we have Λ 1 + − . An easy application of the ⩾ ̃i = ( ) ( i ) Vandermonde convolution− shows that i i i 1 Λ̃i 1 − Λj. = jQ1(− ) i j = − Then Λ F for all i N. Since the coordinate matrix of the ̃i ∈ ⟨T ⟩ ∈ family Λ0,...,Λ relative to Λ0,...,Λ is the involutory lower (̃ ̃i) ( i) triangular Pascal matrix of order i 1, it follows that is a basis of T̃ F and that χ is an involution. + ⟨T ⟩ We note that χ is an F -algebra automorphism is due to the fact that Riordan’s formula remains true for any negative integer k.

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