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Document downloaded from: http://hdl.handle.net/10251/92205 This paper must be cited as: Lebtahi Ep-Kadi-Hahifi, L.; Romero Martínez, JO.; Thome Coppo, NJ. (2017). Algorithms for solving the inverse problem associated with KAK = A(s+1). Journal of Computational and Applied Mathematics. 309:333-341. doi:10.1016/j.cam.2016.02.055 The final publication is available at http://dx.doi.org/10.1016/j.cam.2016.02.055 Copyright Elsevier Additional Information Algorithms for solving the inverse problem associated with KAK = As+1 Leila Lebtahi∗ Oscar´ Romeroy N´estor Thomez Abstract In previous papers, the authors introduced and characterized a class of matrices called K; s + 1 -potent. Also, they established a method to construct thesef matrices.g The purpose of this paper is to solve the associated inverse problem. Several algorithms are developed in order to find all involutory matrices K satisfying KAs+1K = A for n n a given matrix A C × and a given natural number s. The cases s = 0 and s 1 are2 separately studied since they produce different situations. In≥ addition, some examples are presented showing the numerical performance of the methods. Keywords: Involutory matrix; K; s + 1 -potent matrix; Spectrum; Al- gorithm; Inverse problem f g MSC: 15A24, 15A29 1 Introduction n n 2 Let K C × be an involutory matrix, that is K = In, where In denotes the iden2tity matrix of size n n. In [11], the authors introduced and charac- terized a new kind of matrices×called K; s+1 -potent where K is involutory. f g ∗Universidad Internacional de La Rioja. Logrono,~ Spain. E{mail address: [email protected]. yDepartamento de Comunicaciones. Universitat Polit`ecnica de Val`encia. E{46022 Va- lencia, Spain. E{mail address: [email protected]. zInstituto Universitario de Matem´atica Multidisciplinar. Universitat Polit`ecnica de Val`encia. E{46022 Valencia, Spain. E{mail address: [email protected]. 1 n n We recall that for an involutory matrix K C × and s 0; 1; 2; : : : , a n n 2 s+1 2 f g matrix A C × is called K; s + 1 -potent if KA K = A: These matri- ces generalize2 all the followingf classesg of matrices: s + 1 -potent matrices, periodic matrices, idempotent matrices, involutory matrices,f g centrosymmet- ric matrices, mirrorsymmetric matrices, 2 2 circulant matrices, etc. Some related classes of matrices are studied in [3×, 6, 17, 18, 20, 21, 22]. We emphasize that the role of centrosymmetric matrices is very important in different technical areas. We can mention among them antenna theory, pattern recognition, vibration in structures, electrical networks and quan- tum physics (see [1, 5, 8, 9, 19, 23, 25]). It is observed that the computa- tional complexity of various algorithms is reduced taking advantage of the structure of these matrices. Also, mirror-symmetric matrices have important applications in studying odd/even-mode decomposition of symmetric mul- ticonductor transmission lines [15]. Additional applications of K; s + 1 - potent matrices are related to the calculation of high powers off matrices,g such as those needed in Markov chains and Graph Theory. Allowing nega- tive values for s, Wikramaratna studied in [26] a new type of matrices for generating pseudo-random numbers. Inspired by this idea, another applica- tion, in image processing, has been considered in [14] where algorithms for image blurring/deblurring are designed. The advantage of this method is to avoid the computation of inverses of matrices and it can be applied, for instance, to protect a part of an image. The class of K; s+1 -potent matrices is linked to other kind of matrices such as s + 1 -generalizedf g projectors, K -Hermitian matrices, normal ma- trices, Hamiltonianf g matrices, etc. [12].f Moreog ver, some related results are given in [4] from an algebraic point of view. Furthermore, in [13] the authors developed an algorithm to construct the matrices in this class. This problem is called the direct problem. The aim of this paper is to solve the inverse problem, that is, to find all the involutory matrices K for which a given matrix A is K; s + 1 -potent. For this purpose, several algorithms will be developed. f The s g= 0 and s 1 cases are separately studied since they produce different situations. In addition,≥ some examples are presented showing the numerical performance of the methods. In what follows, we will need the spectral theorem: n n Theorem 1 ([2]) Let A C × with k distinct eigenvalues λ1; : : : ; λk. Then the matrix A is diagonalizable2 if and only if there are disjoint projectors 2 P1; P2; : : : ; Pk, (i.e., PiPj = δijPi for i; j 1; 2; : : : ; k ) such that A = k k 2 f g j=1 λjPj and In = j=1 Pj. th P For a positive integerP k, let Ωk be the set of all k roots of unity. If we define ! = e2πi=k then Ω = !1; !2; : : : ; !k . The elements of Ω will always k f g k be assumed to be listed in this order. Define Λk = 0 Ωk = λ0; λ1; : : : ; λk j f g[ f g so that λ0 = 0, and λj = ! for 1 j k. Let N = 0; 1; 2; : : : ; (s + 1)2 ≤ 2 ≤for an integer s 1. In [11], it was s f − g ≥ proved that the function ' : Ns Ns defined by '(j) j(s + 1) [mod ((s + 1)2 1)] is a permutation. Moreo!ver, we notice that ' is≡an involution. It was also−shown that the eigenvalues of a K; s + 1 -potent matrix A are included f g in the set Λ(s+1)2 1 and such a matrix A has associated certain projectors. − Specifically, we will consider matrices Pj's satisfying the relations KPjK = P'(j) and KP(s+1)2 1K = P(s+1)2 1; (1) − − for j Ns, where P0; P1; : : : ; P(s+1)2 1 are the eigenprojectors given in The- orem 21. For simplicity in the notation,− we are assuming that all of these projectors Pj's are spectral projectors for A. However, for any specific spec- tral decomposition of A we must consider the (unique) spectral projectors needed for that decomposition. The designed algorithms compute these spe- cific eigenprojectors and, in Section 5, we show some examples with their adequate spectral projectors. Those examples also illustrate all the studied situations throughout the paper. n n Theorem 2 ([11]) Let A C × and s 1 be an integer. Then the follow- ing conditions are equivalent:2 ≥ (a) A is K; s + 1 -potent. f g (b) A is diagonalizable, σ(A) Λ(s+1)2 1, and the Pj's satisfy condition (1), where σ(A) denotes the sp⊆ectrum of− A. (s+1)2 (c) A = A, and the Pj's satisfy condition (1). 2 Obtaining the involutory matrices K for s 1 ≥ It is well known that the Kronecker product is an important tool to solve some matrix problems, as for example the Sylvester and Lyapunov equations. The 3 Kronecker sum, obtained as a sum of two Kronecker products, is applied, for example, to solve the two-dimensional heat equation, to rewrite the Jacobi iteration matrix, etc. [24]. The notation used in this paper refers to the Kronecker product; and X T denotes the transp⊗ ose of the matrix X [10]. For n n n2 1 any matrix X = [xij] C × , let v(X) = [vk] C × be the vector formed by stacking the columns2 of X into a single column2 vector. The expression th [v(X)] (j 1)n+1;:::;(j 1)n+n , for j = 1; : : : ; n, denotes the j column of X. f − − g n n In what follows, we will need the following property: if A C × and n n 2 B C × then 2 A Ker(A) Ker(B) = Ker ; (2) \ B which is also valid for a finite number of matrices of suitable sizes, where Ker(:) denotes the null space of the matrix (:). We recall that when A is a diagonalizable matrix whose distinct eigenval- ues are λ1; λ2; : : : ; λl, the principal idempotents are given by p (A) l P = t where p (η) = (η λ ): (3) t p (λ ) t − i t t i = 1 iY6= t By using the function ' and the projectors introduced in (1), it is possible to construct the matrix T (P0 In) + (In P'(0)) T ⊗ ⊗ − (P1 In) + (In P'(1)) 2 ⊗ . ⊗ − 3 M = . : (4) 6 T 2 7 6 (P(s+1)2 2 In) + (In P'((s+1) 2)) 7 − ⊗ ⊗ − − 6 T 2 7 6 (P(s+1)2 1 In) + (In P(s+1) 1) 7 6 − ⊗ ⊗ − − 7 4 5 For a given positive integer s, the square, complex matrix A is called a 2 potential K; s + 1 -potent matrix if A(s+1) = A, or equivalently, if A is f g diagonalizable and σ(A) is contained in Λ(s+1)2 1. Note that K is completely unspecified here. Also note that it is generally −much easier and faster to test 2 that A(s+1) = A than is to determine the spectrum of A and to determine that A is diagonalizable. Algorithm 1 n n Inputs: An integer s 1, and a matrix A C × for some integer n 2. ≥ 2 ≥ 4 Output: A decision on whether A is potentially K; s+1 -potent or not. f g 2 Step 1 If either A(s+1) = A or, A is diagonalizable and σ(A) ⊆ Λ(s+1)2 1, then \A is potentially K; s + 1 -potent." Go to End. − f g Step 2 \A is not potentially K; s + 1 -potent, and there is no f n n g idempotent matrix K C × such that A is K; s + 1 - potent." 2 f g End The algorithm presented below solves the inverse problem stated in the introduction.
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