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By further generalizing the concept of Hermitian (or normal) and skew-Hermitian splitting for a non-Hermitian and positive-definite matrix, we introduce a new splitting, called positive-definite and skew-Hermitian splitting (PSS), and then establish a class of PSS methods similar to the Hermitian (or normal) and skew-Hermitian splitting (HSS or NSS) method for iteratively solving the positive-definite systems of linear equations. Theoretical analysis shows that the PSS method converges unconditionally to the exact solution of the linear system, with the upper bound of its convergence factor dependent only on the spectrum of the positive-definite splitting matrix and independent of the spectrum of the skew-Hermitian splitting matrix as well as the eigenvectors of all matrices involved. When we specialize the PSS to block triangular (or triangular) and skew-Hermitian splitting (BTSS or TSS), the PSS method naturally leads to a BTSS or TSS iteration method, which may be more practical and efficient than the HSS and NSS iteration methods. Applications of the BTSS method to the linear systems of block two-by-two structures are discussed in detail. Numerical experiments further show the effectiveness of our new methods.

The purpose of this paper is to give conditions for the existence and uniqueness of the infinite matrix factorization LU.

As structural engineers move further into the age of digital computation and rely more heavily on computers to solve problems, it remains paramount that they understand the basic mathematics and engineering principles used to design and analyze building structures. The analysis of complex structural systems involves the knowledge of science, technology, engineering, and math to design and develop efficient and economical buildings and other structures. The link between the basic concepts and application to real world problems is one of the most challenging learning endeavors that structural engineers face. A thorough understanding of the analysis procedures should lead to successful structures.

This computationally oriented book describes and explains the mathematical relationships among matrices, moments, orthogonal polynomials, quadrature rules, and the Lanczos and conjugate gradient algorithms. The book bridges different mathematical areas to obtain algorithms to estimate bilinear forms involving two vectors and a function of the matrix. The first part of the book provides the necessary mathematical background and explains the theory. The second part describes the applications and gives numerical examples of the algorithms and techniques developed in the first part. Applications addressed in the book include computing elements of functions of matrices; obtaining estimates of the error norm in iterative methods for solving linear systems and computing parameters in least squares and total least squares; and solving ill-posed problems using Tikhonov regularization. This book will interest researchers in numerical linear algebra and matrix computations, as well as scientists and engineers working on problems involving computation of bilinear forms.

Given a unital C*-algebra A, let Mm×n (A) be the set of all m×n matrices algebra over A and ( M n ( A ) ) 1 be the closed unit ball of Mn×n (A). Let x = ( a b 0 v ) ∈ ( M m + n ( A ) ) 1 be determined by a ∈ Mm×m (A), b ∈ Mm×n (A) and c ∈ Mn×n (A). Some characterizations are given such that the above upper triangular matrix x is an extreme point of ( M n ( A ) ) 1 and Xm,n (A) respectively, where Xm,n (A) is the subset of ( M n ( A ) ) 1 consisting of all upper triangular matrices.

The expressions for the Drazin inverse of two kinds of anti-triangular block matrices are developed under new and weaker assumptions relative to those already used recently in this subject. Applying our results concerning the Drazin inverse and anti-triangular block matrices, we propose some characterizations and representations of the Drazin inverse of a 2 × 2 block matrix. In this way, we expand some notable achievements in characterizing and representing generalized inverses of partitioned matrices.

This paper is devoted to studying the Drazin inverse of certain structured matrices under newly introduced restrictive conditions. Specifically, we focus on the Drazin inverse of two types of anti-triangular block matrices. Moreover, several special cases of these results are also discussed. New representations for the Drazin inverse of an arbitrary block matrix are provided under certain conditions, extending recent results in the literature. Furthermore, a numerical example is presented to illustrate the theoretical findings.

The possible instability of partial indices is one of the important constraints in the creation of approximate methods for the factorization of matrix functions. This paper is devoted to a study of a specific class of triangular matrix functions given on the unit circle with a stable and unstable set of partial indices. Exact conditions are derived that guarantee a preservation of the unstable set of partial indices during a perturbation of a matrix within the class. Thus, even in this probably simplest of cases, when the factorization technique is well developed, the structure of the parametric space (guiding the types of matrix perturbations) is non-trivial.