The goal of this paper is to suggest and establish an efficient iterative method based on matrix by matrix multiplications for finding the approximate inverse of nonsingular square matrices. We then analytically extend this proposed method so as to compute the Moore-Penrose generalized inverse of a non-square matrix. A theoretical analysis has been employed to compare the computational efficien...

Definition 1.1. A matrix A ∈ Zm×n is totally unimodular if the determinant of every square submatrix B ∈ Zk×k equals either −1, 0, or +1. Alternatively, by Cramer’s rule, A ∈ Zm×n is totally unimodular if every nonsingular submatrix B ∈ Zk×k has an integral inverse B−1 ∈ Zk×k. Recall: B−1 = 1 det B B ∗, where B∗ is the adjugate matrix (transpose of the matrix of cofactors) of B. One important c...

We consider the SIMPLE preconditioning for block two-by-two generalized saddle point problems; this is the general nonsymmetric, nonsingular case where the 1,2 block needs not to equal the transposed 2,1 block, and the 2,2 block may not be zero. The eigenvalue analysis of the SIMPLE preconditioned matrix is presented. The relationship between the two different formulations spectrum of the SIMPL...

Associated with any matrix, there are four fundamental subspaces: the column space, row space, (right) null space, and left null space. We describe a single computation that makes readily apparent bases for all four of these subspaces. Proofs of these results rely only on matrix algebra, not properties of dimension. A corollary is the equality of column rank and row rank. Bringing a matrix to r...

In this paper, fast algorithms for solving RFPrLR circulant linear systems are presented by the fast algorithm for computing polynomials. The unique solution is obtained when the RFPrLR circulant matrix over the complex field C is nonsingular, and the special solution and general solution are obtained when the RFPrLR circulant matrix over the complex field C is singular. The extended algorithms...

We present an on-shell scheme to renormalize the Cabibbo-Kobayashi-Maskawa (CKM) matrix. It is based on a novel procedure to separate the external-leg mixing corrections into gauge-independent self-mass and gauge-dependent wave function renormalization contributions, and to implement the on-shell renormalization of the former with nondiagonal mass counterterm matrices. Diagonalization of the co...

Conditioning of a nonsingular matrix subspace is addressed in terms of its best conditioned elements. Computationally the problem is seemingly challenging. By associating with the task an intersection problem with unitary matrices leads to a more accessible approach. A resulting matrix nearness problem can be viewed to generalize the so-called Löwdin problem in quantum chemistry. For critical p...

For matrix games we study how small nonzero probability must be used in optimal strategies. We show that for n × n win-lose-draw games (i.e. (−1, 0, 1) matrix games) nonzero probabilities smaller than n are never needed. We also construct an explicit n × n win-lose game such that the unique optimal strategy uses a nonzero probability as small as n. This is done by constructing an explicit (−1, ...

The paper discusses possible forms of square matrices whose rows are pairwise orthogonal and the same is true their columns. This discussion applied to problem finding conditions under which a nonsingular binormal matrix unitoid. A said be if AA∗ A∗A commute. unitoid it can brought diagonal form by (Hermitian) congruence.

Let $G$ be an infinite-dimensional representation-theoretic Kac--Moody group associated to a nonsingular symmetrizable generalized Cartan matrix. We consider Eisenstein series on induced from unramified cusp forms finite-dimensional Levi subgroups of maximal parabolic subgroups. Under natural condition subgroups, we prove that the cuspidal are entire full complex plane.