Matrices A such that RA = As+1R when Rk = I Leila Lebtahi Jeffrey Stuarty Néstor Thome James R. Weaverz Abstract n n s+1 This paper examines matrices A C such that RA = A R where Rk = I, the identity matrix, and where2 s and k are nonnegative integers with k 2. Spectral theory is used to characterize these matrices. The cases s ≥= 0 and s 1 are considered separately since they are analyzed by different techniques.≥ Keywords: Potent matrix; idempotent matrix; spectrum; Jordan form; invo- lutory matrix. AMS subject classification: Primary: 15A18; Secondary: 15A09, 15A21 1 Introduction and Preliminaries Let R1 be the square matrix with ones on the cross diagonal and zeros else- where; note that R1 is often called the centrosymmetric permutation matrix.A matrix A1 that commutes with R1 is called a centrosymmetric matrix [12]. Any 2 square matrix R2 satisfying R2 = I, where I is the identity matrix, is called an involution or an involutory matrix. The real eigenvalues of nonnegative matri- ces that commute with a real involution were studied in [13]. It is well-known that if P is a permutation matrix, then P k = I for some positive integer k. Matrices that commute with a permutation matrix P were studied in [8]. A well-known and important class of matrices that commute with a permutation matrix are the circulant matrices [3, 6], consisting of all matrices that commute with R3, where R3 is the irreducible permutation matrix with ones on the first superdiagonal, a one in the lower left-hand corner, and zeros elsewhere. If A is n 1 an n n circulant matrix, then R3A = AR3 can be expressed as R3AR3 = A n since R = In, the n n identity matrix. 3 Instituto Universitario de Matemática Multidiscilpinar. Universitat Politècnica de Valèn- cia. E—46022 Valencia, Spain. E-mail: {leilebep,njthome}@mat.upv.es. This work has been partially supported by the Ministry of Education of Spain (Grant DGI MTM2010-18228). yDepartment of Mathematics, Pacific Lutheran University, Tacoma, WA 98447 USA. E- mail: [email protected]. zDepartment of Mathematics and Statistics, University of West Florida, Pensacola, FL 32514, USA. E-mail: [email protected]. 1 n n k A matrix R C such that R = In for some positive integer k with k 2 is called a k -involutory2 matrix [10, 11]. Throughout this paper, all matrices≥ f g R will be k -involutory. It is clear that when k = 2, such an R is either In, or else a nontrivialf g involution. Also, k = n is the smallest positive integer for± which R3 is k -involutory, and this guarantees that there are nontrivial, nondiagonal k -involutoryf g matrices for all integers k and n with n 2 and 2 k n. The f g 2i ≥ ≤ ≤ matrix exp( k ) In is k -involutory for all positive integers n and all integers k 2, and, thus, it shouldf g be clear that there are k -involutory matrices for ≥ f g which k > n must occur. Finally, we always assume that R = In, and hence, if k 6 R = In, then k 2. This paper is≥ focused on the study of the R, s + 1, k -potent matrices. A n n f g s+1 matrix A C is called an R, s + 1, k -potent matrix if RA = A R for some nonnegative2 integer s andf some k -involutoryg matrix R. Note that the cases, k = 2 and s 1, and k 2 andf gs = 0, have already been analyzed in [7, 14], respectively.≥ Spectral properties≥ of matrices related to the R, s + 1, k - potent matrices are presented in [4, 9]. Other similar classes of matricesf andg their spectral properties have been studied in [5, 9, 10, 11]. In this paper characterizations of R, s + 1, k -potent matrices are given, with the cases s 1 and s = 0 treatedf separately.g In the first case, the concept of t+1 -group involutory≥ matrix will be used. These matrices were introduced inf [2] forg t = 2, and the definition can be extended for any integer t > 2 as n n follows: A matrix A C is called a t + 1 -group involutory matrix if # t 1 # 2 f g A = A , where A denotes the group inverse of A. We recall that the group inverse of a square matrix A is the only matrix A# (when it exists) satisfying: AA#A = A, A#AA# = A#, AA# = A#A. Moreover, A# exists if and only if rank(A2) = rank(A) [1]. 2 Main results Clearly, In and n n zero matrix O are always R, s + 1, k -potent matrices. For any given positive integers n, s and k (with k f 2), and forg any given n n k -involutory matrix R, there exists a nontrivial≥ R, s + 1, k -potent matrix. f g th f g Consider A = !In where ! is a primitive s root of unity. Note that when s = 0, A = R is an R, 1, k -potent matrix that is nontrivial when R is nontrivial. The questionf arisingg in this paper follows from the observation that if A n n (s+1)k 2 C is an R, s + 1, k -potent matrix, then A = A. To see this, note that from RAf = As+1Rg , it follows that R2A = R(AAsR) = As+1RAsR = s+1 s+1 s 1 (s+1)(s+1) 2 k (s+1)k k A A RA R = = A R , and similarly, R A = A R . (The equality in the observation··· is uninformative when s = 0; the s = 0 case k will be addressed in Subsection 2.2.) The necessity of A(s+1) = A is clear, but is this condition suffi cient to guarantee that a matrix A is an R, s+1, k -potent matrix for an arbitrary k -involution R? Not surprisingly,f since R doesg not appear in the equality, thef g condition is not suffi cient as the following example 2 demonstrates: 2i A = exp I2,R = diag(i, 1), s = 1, k = 4. 3 Consequently, we seek a complementary condition that in conjunction with k A(s+1) = A implies A is an R, s + 1, k -potent matrix. f g 2.1 The case s 1 ≥ k Assume that A is an R, s + 1, k -potent matrix. Let ns = (s + 1) 1. Since k f g k A(s+1) = A, the polynomial t(s+1) t, whose roots all have multiplicity 1, is divisible by the minimal polynomial of A. Thus, A is diagonalizable with 1 2 ns 1 ns 2i spectrum (A) 0 ! ,! ,...,! ,! = 1 where ! := exp . f g [ f g ns Hence, the spectral theorem [1] assures that there exist disjoint projectors P0,P1,P2,...,Pns 1,Pns such that ns ns j A = ! Pj and Pj = In, (1) j=1 j=0 X X j0 where Pj = O if there exists j0 1, 2, . , ns such that ! / (A) and 0 2 f g 2 moreover that P0 = O when 0 / (A). Pre-multiplying the previous2 expressions given in (1) by the matrix R and 1 post-multiplying by R gives ns 1 j 1 RAR = ! RPjR j=1 X and ns 1 RPjR = In. (2) j=0 X 1 It is clear that the nonzero RPjR are disjoint projectors for each j = 0, 1, . , ns. From (1), ns s+1 j(s+1) A = ! Pj j=1 X because the nonzero Pj are disjoint projectors. Let S = 1, 2, . , ns 1 . Now consider ' : S 0 S 0 as the f g [ f g ! [ f g function defined by '(j) = bj, where bj is the smallest nonnegative integer such that bj j(s + 1) [mod ns]. Then ' is a bijection [7]. It follows that ns 1 s+1 '(j) A = ! Pj + Pns j=1 X 3 and since A is an R, s + 1, k -potent matrix, f g s+1 1 A = RAR . Hence, ns 1 ns 1 i 1 1 '(j) ! RPiR + RPnx R = ! Pj + Pns . i=1 j=1 X X Since ' is a bijection, for each i S, there exists a unique j S such that i = '(j). From the uniqueness of the2 spectral decomposition, it follows2 that for every i S, there exists a unique j S such that 2 2 1 1 RPiR = RP'(j) R = Pj. (3) It is clear that uniqueness also implies that 1 RPns R = Pns . (4) Finally, from (1) ns P0 = In Pj. j=1 X Taking into account (2) and the definition of the bijection ', 1 RP0R = P0 (5) because of the uniqueness of the spectral decomposition. Observe that in the j0 case where there exists j0 S such that ! / (A), it has been indicated that 2 12 Pj0 = O. In this situation, P'(j0) = RPj0 R = O is also true. k Conversely, assuming A(s+1) = A and that the relationships on the projec- tors obtained in (3), (4), and (5) hold, we can consider ns j A = ! Pj (6) j=1 X s+1 1 It is now easy to check that A = RAR . The matrices Pj’ssatisfying relations (3), (4), and (5) where P0,P1,...,Pns are the projectors appearing in the spectral decomposition of A associated to the eigenvalues 1 nx 1 0,! ,...,! , 1, are said to satisfy condition ( ). Then, the complementary condition we were looking for is condition ( ). P These results are summarizedP in what follows. Before that, note k rank(A) = rank(A(s+1) ) rank(A2) rank(A) ≤ ≤ 4 k when A(s+1) = A.