In this paper we parallelly build up the theories of normed linear spaces and of linear spaces with indefinite metric, called also Minkowski spaces for finite dimensions in the literature. In the first part of this paper we collect the common properties of the semi-and indefinite-inner-products and define the semi-indefinite-inner-product and the corresponding structure, the semi-indefinite-inn...

In this paper, we give the generalized alternating twostage method in which the inner iterations are accomplished by a generalized alternating method. And we present convergence results of the method for solving nonsingular linear systems when the coefficient matrix of the linear system is a monotone matrix or an H-matrix. Keywords—generalized alternating two-stage method, linear system, conver...

There are also four appendices. Let K be a field of characteristic 0, and let C be a commutative K-algebra. which makes C into a Lie algebra, and is a biderivation (i.e. a derivation in each argument). The pair C, {−, −} is called a Poisson algebra. Poisson brackets arise in several ways. Example 1.1. Classical Hamiltonian mechanics. Here K = R, X is an even dimensional differentiable manifold ...

Let T and S be a subspace of C and C, respectively. Then for A ∈ Cm×n satisfied AT ⊕ S = C, the generalized inverse A T,S is given by A (2) T,S = (PS⊥APT ) †. In this paper, a finite formulae is presented to compute generalized inverse A T,S under the concept of restricted inner product, which defined as < A,B >T,S=< PS⊥APT , B > for the A,B ∈ Cm×n. By this iterative method, when taken the init...

In this paper we consider the numerical solution of large-scale projected generalized continuous-time and discrete-time Sylvester equations with low-rank right-hand sides. First, we present the results on the sufficient conditions for the existence, uniqueness, and analytic formula of the solutions of these equations. Second, we review the low-rank alternating direction implicit method and the ...