In this paper, only simple graphs are considered. A graphG is nonsingular if its adjacency matrix A(G) is nonsingular. A nonsingular graph G satisfies reciprocal eigenvalue property (property R) if the reciprocal of each eigenvalue of the adjacency matrix A(G) is also an eigenvalue of A(G) and G satisfies strong reciprocal eigenvalue property (property SR) if the reciprocal of each eigenvalue o...

A block graph is a graph in which every block is a complete graph (clique). Let G be a block graph and let A(G) be its (0,1)-adjacency matrix. Graph G is called nonsingular (singular) if A(G) is nonsingular (singular). Characterizing nonsingular block graphs, other than trees, is an interesting open problem. In this article, we give some classes of nonsingular and singular block graphs and rela...

The matrix inversion plays a signifcant role in engineering and sciences. Any nonsingular square matrix has a unique inverse which can readily be evaluated via numerical techniques such as direct methods, decomposition scheme, iterative methods, etc. In this research article, first of all an algorithm which has fourth order rate of convergency with conditional stability will be proposed. ...

A square real matrix is sign-nonsingular if it is forced to be nonsingular by its pattern of zero, negative, and positive entries. We give structural characterizations of sign-nonsingular matrices, digraphs with no even length dicycles, and square nonnegative real matrices whose permanent and determinant are equal. The structural characterizations, which are topological in nature, imply polynom...

In this paper, we investigate the reduced form of circulant matrices and we show that the problem of computing the q-th roots of a nonsingular circulant matrix A can be reduced to that of computing the q-th roots of two half size matrices B - C and B + C.

In this paper we present different Monte Carlo algorithms to obtain inverse matrix. Here, we first determine two algorithms and describe their restrictions to be applicable for computing the inversion of a nonsingular matrix. Then, we establish an efficient algorithm to obtain approximate inversion of an arbitrary nonsingular matrix. This algorithm obtain an accurate inversion when the other tw...

23.1 Let A be a nonsingular square matrix and let A = QR and A∗A = U∗U be QR and cholesky factorizations, respectively, with the usual normalization rjj, ujj > 0. Is it true or false that R = U? Solution: It is true. Since A is nonsingular, it will have a unique QR factorization with rjj > 0. Then we have A∗A = R∗Q∗QR = R∗R because Q is a unitary matrix. On the other hand, it is easy to see tha...

A graph G is said to be nonsingular (resp., singular) if its adjacency matrix A(G) is nonsingular (resp., singular). The inverse of a nonsingular graph G is the unique weighted graph whose adjacency matrix is similar to the inverse of the adjacency matrix A(G) via a diagonal matrix of ±1s. Consider connected bipartite graphs with unique perfect matchings such that the graph obtained by contract...

In the linear system Ax = b the points x are sometimes constrained to lie in a given subspace S of column space of A. Drazin inverse for any singular or nonsingular matrix, exist and is unique. In this paper, the singular consistent or inconsistent constrained linear systems are introduced and the effect of Drazin inverse in solving such systems is investigated. Constrained linear system arise ...

To a given nonsingular triangular matrix A with entries from a ring, we associate a weighted bipartite graph G(A) and give a combinatorial description of the inverse of A by employing paths in G(A). Under a certain condition, nonsingular triangular matrices A such that A and A−1 have the same zero-nonzero pattern are characterized. A combinatorial construction is given to construct outer invers...