Abstract: The considerations of the present paper were inspired by Baksalary [O.M. Baksalary, Idempotency of linear combinations of three idempotent matrices, two of which are disjoint, Linear Algebra Appl. 388 (2004) 67–78] who characterized all situations in which a linear combination P=c_1P_1+c_2P_2+c_3P_3, with c_i, i=1,2,3, being nonzero complex scalars and P_i, i=1,2,3, being nonzero complex idempotent matrices such that two of them, P_1 and P_2 say, are disjoint, i.e., satisfy condition P_1P_2=0=P_2P_1, is an idempotent matrix. In the present paper, by utilizing different formalism than the one used by Baksalary, the results given in the above mentioned paper are generalized by weakening the assumption expressing the disjointness of P_1 and P_2 to the commutativity condition P_1P_2=P_2P_1.
Abstract: In this article, two facts related to the generalized Schur complement are studied. The first one is to find necessary and sufficient conditions to characterize when the group inverse of a partitioned matrix can be expressed in the Schur form. The other one is to develop a formula for any power of the generalized Schur complement of an idempotent partitioned matrix and then to characterize when this generalized Schur complement is a (k+1)-potent matrix. In addition, some spectral theory related to this complement is analyzed.
Abstract: In this paper we deal with two problems related to {k}-group periodic matrices (i.e., $A^{\#} = A^{k-1}$, where $A^{\#}$ is the group inverse of a matrix A). First, we give different characterizations of {k}-group periodic matrices. Later, we present characterizations of the {k}-group periodic matrices for linear combinations of projectors. This work extends some well-known results in the literature.
Abstract: In this work we characterize the matrices A with the following property: for each ε>0 there exists a natural k such that \|A^k-I\| < ε for a given matrix norm. This characterization is applied to the theory of unitary, Hermitian nonnegative, positive and stochastic matrices.
Abstract: The purpose of this note is to present a simple proof (without using the Gauss egregium theorem) of the following fact: To make a length preserving projection of the Earth is impossible.
Abstract: This paper deals with idempotent matrices (i.e., A^2 = A) and t-potent matrices (i.e., B^t = B). When both matrices commute, we derive a list of all complex numbers c1 and c2 such that c1A + c2B is an idempotent matrix. In addition, the real case is also analyzed.
Abstract: We extend generalized projectors (introduced by Groß and Trenkler [Generalized and hypergeneralized projectors, Linear Algebra Appl. 264 (1997) 463–474]) to k-generalized projectors and we characterize them obtaining results in the aforesaid paper as a consequence. Moreover, we list all situations when a linear combination of commuting k-generalized projectors is a k-generalized projector. The method for obtaining this result permits to give a revisited version of the main result by Baksalary and Baksalary [On linear combinations of generalized projectors, Linear Algebra Appl. 388 (2004) 17–24]. In addition, the case of orthogonal projectors is also analyzed.
Abstract: This work introduces an application of differential geometry to cartography. The mathematical aspects of some geographical projections of Earth surface are revealed together with some of its more important properties.
