several representations of the generalized drazin inverse of an anti-triangular block matrix in banach algebra are given in terms of the generalized banachiewicz--schur form.

For little q-Jacobi polynomials, q-Hahn polynomials and big q-Jacobi polynomials we give particular q-hypergeometric series representations in which the termwise q = 0 limit can be taken. When rewritten in matrix form, these series representations can be viewed as decompositions into a lower triangular matrix times upper triangular matrix. We develop a general theory of such decompositions rela...

We prove a lower bound of 52n 2 3n for the rank of n n–matrix multiplication over an arbitrary field. Similar bounds hold for the rank of the multiplication in noncommutative division algebras and for the multiplication of upper triangular matrices.

In this paper, we study the relations between the Bell matrix and the Fibonacci matrix, which provide a unified approach to some lower triangular matrices, such as the Stirling matrices of both kinds, the Lah matrix, and the generalized Pascal matrix. To make the results more general, the discussion is also extended to the generalized Fibonacci numbers and the corresponding matrix. Moreover, ba...

Nakajima conjectured in [N] that the transition matrix between certain PBW-basis constructed in [LXZ] and the affine canonical basis is upper triangular with the diagonal entries equal to one and the upper diagonal entries in vZ[v]. In this paper, we show that the transition matrix between the PBW-basis in [LXZ] and the canonical basis is upper triangular with the diagonal entries equal to one ...

Let k be a fixed field of arbitrary characteristic and let ? basic connected Nakayama k-algebra without simple projective modules. In this article we prove that if V is an indecomposable finitely generated Gorenstein-projective left ?-module, then the versal deformation ring R(?,V) (in sense F. M. Bleher author) universal stable after taking syzygies. We also following result. ?=(?B0?) triangul...

We study the neutrino mixing matrix (the MNS matrix) in the seesaw model. By assuming a large mass hierarchy for the heavy right-handed Majorana mass, we show that, in the diagonal Majorana base, the MNS matrix is determined by a unitary matrix, S, which transforms the neutrino Yukawa matrix, yν , into a triangular form, y△. The mixing matrix of light leptons is VKMS ′∗ , where VKM ≡ VLe VLν an...

in this paper, the notions of $(t,s)$-composition matrix and$(t,s)$-interval-valued intuitionistic fuzzy equivalence matrix areintroduced where $(t,s)$ is a dual pair of triangular module. theyare the generalization of composition matrix and interval-valuedintuitionistic fuzzy equivalence matrix. furthermore, theirproperties and characterizations are presented. then a new methodbased on $tilde{...