On the spectra of some combinations of two generalized quadratic matrices

Abstract

Elsevier Petik, T.; Ozdemir, H.; Benítez López, J. (2015). On the spectra of some combinations of two generalized quadratic matrices. Applied Mathematics and Computation. 978-990. doi:10.1016/j.amc.2015.06 Abstract Let A and B be two generalized quadratic matrices with respect to idempotent matrices P and Q, respectively, such that (A − αP )(A − βP ) = 0, AP = P A = A, (B − γQ)(B − δQ) = 0, BQ = QB = B, P Q = QP , AB = BA, and (A + B)(αβP − γδQ) = (αβP − γδQ)(A + B) with α, β, γ, δ ∈ . Let A + B be diagonalizable. The relations between the spectrum of the matrix A + B and the spectra of some matrices produced from A and B are considered. Moreover, some results on the spectrum of the matrix A + B are obtained when A + B is not diagonalizable. Finally, some results and examples illustrating the applications of the results in the work are given.

On involutiveness of linear combinations of a quadratic matrix and an arbitrary matrix

“…In [1,2,4,7,13], it has been characterized the involutiveness of the form aA + bB when a, b ∈ C and A, B are special types of matrices. Moreover, there are a lot of studies related to the linear combinations including involutive matrices [4,7,9,14] and quadratic, generalized quadratic matrices [2,3,5,6,8,10,11]. These special types of matrices have applications to digital image encryption (for example, [12]).…”