-CANONICAL FORMS AND DRAZIN INVERSES P M. MOUC¸OUF k Abstract. In this paper, P-canonical forms of (A )k (or sim- ply of the matrix A) are defined and some of their properties are proved. It is also shown how we can deduce from them many inter- esting 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 nonsingu- lar, 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 recur- rence sequences, Binomial coefficients, Drazin inverses. 1 2 M. MOUC¸OUF 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 ob- tained [3, 7, 8, 14, 22] and a variety of direct and iterative methods have been developed for computation of this type of generalized in- verse [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 re- currence 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. MOUC¸OUF 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.