Generalized centro-invertible matrices with applications

Abstract

Centro-invertible matrices were introduced by R.S. Wikramaratna in 2008. From an involutory matrix, we introduce generalized centro-invertible matrices and apply them to the modular arithmetic case. Specifically, algorithms for image blurring/deblurring are designed by means of generalized centro-invertible matrices. In addition, we establish that every pair of sets of generalized centro-invertible matrices corresponding to two fixed involutory matrices have the same number of elements.

Further results on generalized centro-invertible matrices

“…We recall that for a given involutory matrix K ∈ C n×n , a matrix A ∈ C n×n is called generalized centro-invertible if KAK = A −1 [11]. It is clear that this class of matrices is a subclass of nonsingular matrices with determinant ±1.…”

“…In addition, some algorithms were given to construct those matrices. Lastly, in [11], the case for nonsingular matrices was analyzed. Some applications were given for the study of blurring/deblurring images where nonsingularity is crucial.…”

“…In addition, a more general situation where the equation KA = A s+1 K is studied for involutory matrices K has been analyzed in [8,10] for nonnegative integer values of s. They are called {K, s + 1}-potent matrices and when s = 0 it is called a {K}-centrosymmetric matrix. Complementing this study, the case s = −2, that corresponds to generalized centro-invertible matrices, has been considered in [11]. In addition, some applications for image blurring/deblurring were developed.…”

Spectral study of {R,s+1,k}- and {RCatral1, Lebtahi2, Stuart3 et al. 2020Journal of Computational and Applied Mathematics2020View full textAdd to dashboardCite

“…Algorithms for constructing matrices in this class were presented in [21] for k = 2, and the inverse problem was solved in [22] where several algorithms were designed in order to compute all involutory matrices R such that RA = A s+1 R. With the necessary modifications, an interesting application of a class of matrices related to {R, s + 1, k}-potent matrices was given in [23] to study image processing.…”

Algorithms for solving the inverse problem associated withKAK=As+1

“…Allowing negative values for s, Wikramaratna studied in [24] a new type of matrices for generating pseudo-random numbers. Inspired by this idea, another application, in image processing, has been considered in [12] where algorithms for image blurring/deblurring are designed. The advantage of this method is to avoid the computation of inverses of matrices and it can be applied, for instance, to protect a part of an image.…”