In this paper, using some block-operator matrix techniques, we give the necessary and sufficient conditions for the reverse order law for {1, 2, 3} and {1, 2, 4}−inverses of bounded operators on Hilbert spaces. Furthermore, we present new equivalent conditions for the reverse order law for the Moore-Penrose inverse. AMS classification: 15A09

The intimate relationship between the Penrose and the Taylor-Socolar tilings is studied, within both the context of double hexagon tiles and the algebraic context of hierarchical inverse sequences of triangular lattices. This unified approach produces both types of tilings together, clarifies their relationship and offers straightforward proofs of their basic properties.

In this paper, necessary and sufficient conditions are given for the existence of Moore-Penrose inverse a product two matrices in an indefinite inner space (IIPS) which reverse order law holds good. Rank equivalence formulas with respect to IIPS provided open problem is at end.

Under certain conditions, we prove that the Moore-Penrose inverse of a sum operators is inverses. From this, derive expressions and characterizations for an operator are useful its computation. We give formulations them finite matrices study circulant distance graphs.

By using the complex representation of quaternion matrices, the Moore–Penrose generalized inverse and the Kronecker product of matrices, the expressions of the least squares η-Hermitian solution with the least norm and the expressions of the least squares η-anti-Hermitian solution with the least norm are derived for the matrix equation AXB+CXD = E over quaternions.

A connected signed graph Ġ, where all blocks of it are positive cliques or negative (of possibly varying sizes), is called a block graph. Let A, N and D̃ be adjacency, net Laplacian distance matrices graph, respectively. In this paper the formulas for determinant were given firstly. Then inverse (resp. Moore-Penrose inverse) obtained if nonsingular singular), which sum Laplacian-like matrix at m...

Generalized inverses of tensors play increasingly important roles in computational mathematics and numerical analysis. It is appropriate to develop the theory generalized within algebraic structure a ring. In this paper, we study different over commutative ring noncommutative Several examples are provided support theoretical results. We also propose algorithms for computing inner inverses, Moor...