The first Chevalley-Eilenberg cohomology group of the Lie algebra on the transverse bundle of a decreasing family of foliations

In [L. Lebtahi, Lie algebra on the transverse bundle of a decreasing family of foliations, J. Geom. Phys. 60 (2010), 122-133], we defined the transverse bundle V^k to a decreasing family of k foliations F_i on a manifold M. We have shown that there exists a (1,1) tensor J of V^k such that Jk≠0, J^(k+1) = 0 and we defined by L_J(V^k) the Lie Algebra of vector fields X on V^k such that, for each vector field Y on V^k, [X,JY]=J[X,Y]. In this note, we study the first Chevalley-Eilenberg Cohomology Group, i.e. the quotient space of derivations of L_J(V^k) by the subspace of inner derivations, denoted by H^1(L_J(V^k))

A Note on k-generalized projections

In this note, we investigate characterizations for k-generalized projections (i.e., A^k =A*) on Hilbert spaces. The obtained results generalize those for generalized projections on Hilbert spaces in [Hong-Ke Du, Yuan Li, The spectral characterization of generalized projections, Linear Algebra Appl. 400 (2005) 313-318] and those for matrices in [J. Benítez, N. Thome, Characterizations and linear combinations of k-generalized projectors, Linear Algebra Appl. 410 (2005) 150-159]

Properties of a matrix group associated to a {K,s+1}-potent matrix

In a previous paper, the authors introduced and characterized a new kind of matrices called {K, s+1}-potent. In this paper, an associated group to a {K, s+1}-potent matrix is explicitly constructed and its properties are studied. Moreover, it is shown that the group is a semidirect product of Z_2 acting on Z_(s+1)^2−1. For some values of s, more specifications on the group are derived. In addition, some illustrative examples are given

On a matrix group constructed from an {R,s+1,k}-potent matrix

Let R∈C^(n×n) be a {k}-involutory matrix (that is, R^k=I_n) for some integer k≥2, and let s be a nonnegative integer. A matrix A∈C^(n×n) is called an {R,s+1,k}-potent matrix if A satisfies R A = A^(s+1) R. In this paper, a matrix group corresponding to a fixed {R,s+1,k}-potent matrix is explicitly constructed, and properties of this group are derived and investigated. This group is then reconciled with the classical matrix group G_A that is associated with a generalized group invertible matrix A. Let R∈Cn×n be a {k}-involutory matrix (that is, Rk=In) for some integer k≥2, and let s be a nonnegative integer. A matrix A∈Cn×n is called an {R,s+1,k}-potent matrix if A satisfies RA=As+1R. In this paper, a matrix group corresponding to a fixed {R,s+1,k}-potent matrix is explicitly constructed, and properties of this group are derived and investigated. This group is then reconciled with the classical matrix group GA that is associated with a generalized group invertible matrix A

Algorithms for solving the inverse problem associated with KAK =A^(s+1)

In previous papers, the authors introduced and characterized a class of matrices called {K,s+1}-potent. Also, they established a method to construct these matrices. 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 K A^(s+1) K = A for a given matrix A∈C^(n×n) and a given natural number s. The cases s=0 and s≥ are separately studied since they produce different situations. In addition, some examples are presented showing the numerical performance of the methods

Inverse eigenvalue problem for normal J-hamiltonian matrices

A complex square matrix A is called J-hamiltonian if AJ is hermitian where J is a normal real matrix such that J^2=−I_n. In this paper we solve the problem of finding J-hamiltonian normal solutions for the inverse eigenvalue problem

Lie algebra on the transverse bundle of a decreasing family of foliations

J. Lehmann-Lejeune in [J. Lehmann-Lejeune, Cohomologies sur le fibré transverse à un feuilletage, C.R.A.S. Paris 295 (1982), 495-498] defined on the transverse bundle V to a foliation on a manifold M, a zero-deformable structure J such that J^2 = 0 and for every pair of vector fields X,Y on M: [JX,JY]−J[JX,Y]−J[X,JY]+J^2[X,Y]=0. For every open set Ω of V, J. Lehmann-Lejeune studied the Lie Algebra L_J(Ω) of vector fields X defined on Ω such that the Lie derivative L(X)J is equal to zero i.e., for each vector field Y on Ω: [X,JY]=J[X,Y] and showed that for every vector field X on Ω such that X∈KerJ, we can write X=∑[Y,Z] where ∑ is a finite sum and Y,Z belongs to L_J(Ω)∩(KerJ|Ω). In this note, we study a generalization for a decreasing family of foliations

Matrices A such that A^{s+1}R = RA* with R^k = I

We study matrices A in C^{nxn} such that A^(s+1)R = RA* where R^k = I_n, and s, k are nonnegative integers with k >= 2; such matrices are called {R,s + 1,k; *}-potent matrices. The s = 0 case corresponds to matrices such that A = RA*R^(-1) with R^k = In, and is studied using spectral properties of the matrix R. For s >= 1, various characterizations of the class of {R,s + 1,k, *}-potent matrices and relationships between these matrices and other classes of matrices are presented

Spectral study of {R, s + 1, k}- and {R, s + 1, k, ∗}-potent matrices

The {R, s+1, k}- and {R, s+1, k, ∗}-potent matrices have been studied in several recent papers. We continue these investigations from a spectral point of view. Specifically, a spectral study of {R,s + 1, k}-potent matrices is developed using characterizations involving an associated matrix pencil (A, R). The corresponding spectral study for {R, s+1, k, ∗}-potent matrices involves the pencil (A∗, R). In order to present some properties, the relevance of the projector I −AA# where A# is the group inverse of A is highlighted. In addition, some applications and numerical examples are given, particularly involving Pauli matrices and the quaternions

The diamond partial order in rings

This is an author's accepted manuscript of an article published in " Linear and Multilinear Algebra"; Volume 62, Issue 3, 2014; copyright Taylor & Francis; available online at: http://dx.doi.org/10.1080/03081087.2013.779272In this paper we introduce a new partial order on a ring, namely the diamond partial order. This order is an extension of a partial order defined in a matrix setting in [J.K. Baksalary and J. Hauke, A further algebraic version of Cochran’s theorem and matrix partial orderings, Linear Algebra and its Applications, 127, 157–169, 1990]. We characterize the diamond partial order on rings and study its relationships with other partial orders known in the literature. We also analyse successors, predecessors and maximal elements under the diamond order.The first and third authors have been partially supported by Ministry of Education of Spain, grant DGI MTM2010-18228 and the third one by Universidad Nacional de La Pampa, Facultad de Ingenieria (grant Resol. No 049/11). The second author was financed by FEDER Funds through 'Programa Operacional Factores de Competitividade - COMPETE' and by Portuguese Funds through FCT - 'Fundacao para a Ciencia e a Tecnologia', within the project PEst-C/MAT/UI0013/2011.Lebtahi Ep-Kadi-Hahifi, L.; Patricio, P.; Thome, N. (2014). The diamond partial order in rings. Linear and Multilinear Algebra. 62(3):386-395. https://doi.org/10.1080/03081087.2013.779272386395623Mitra, S. K., & Bhimasankaram, P. (2010). MATRIX PARTIAL ORDERS, SHORTED OPERATORS AND APPLICATIONS. SERIES IN ALGEBRA. doi:10.1142/9789812838452Baksalary, J. K., & Hauke, J. (1990). A further algebraic version of Cochran’s theorem and matrix partial orderings. Linear Algebra and its Applications, 127, 157-169. doi:10.1016/0024-3795(90)90341-9Patrício P, Mendes Araujo C. Moore-Penrose invertibility in involutory rings: the caseaa†=bb†. Linear and Multilinear Algebra. 2010;58:445–452.Blackwood, B., Jain, S. K., Prasad, K. M., & Srivastava, A. K. (2009). Shorted Operators Relative to a Partial Order in a Regular Ring. Communications in Algebra, 37(11), 4141-4152. doi:10.1080/00927870902828629Baksalary, J. K., Baksalary, O. M., & Liu, X. (2003). Further properties of the star, left-star, right-star, and minus partial orderings. Linear Algebra and its Applications, 375, 83-94. doi:10.1016/s0024-3795(03)00609-8Baksalary, J. K., Baksalary, O. M., Liu, X., & Trenkler, G. (2008). Further results on generalized and hypergeneralized projectors. Linear Algebra and its Applications, 429(5-6), 1038-1050. doi:10.1016/j.laa.2007.03.029Hauke, J., Markiewicz, A., & Szulc, T. (2001). Inter- and extrapolatory properties of matrix partial orderings. Linear Algebra and its Applications, 332-334, 437-445. doi:10.1016/s0024-3795(01)00294-4Mosić, D., & Djordjević, D. S. (2012). Some results on the reverse order law in rings with involution. Aequationes mathematicae, 83(3), 271-282. doi:10.1007/s00010-012-0125-2Mosić, D., & Djordjević, D. S. (2011). Further results on the reverse order law for the Moore–Penrose inverse in rings with involution. Applied Mathematics and Computation, 218(4), 1478-1483. doi:10.1016/j.amc.2011.06.040Tošić, M., & Cvetković-Ilić, D. S. (2012). Invertibility of a linear combination of two matrices and partial orderings. Applied Mathematics and Computation, 218(9), 4651-4657. doi:10.1016/j.amc.2011.10.05