The sign-real and the sign-complex spectral radius, also called the generalized spectral radius, proved to be an interesting generalization of the classical Perron-Frobenius theory (for nonnegative matrices) to general real and to general complex matrices, respectively. Especially the generalization of the well-known Collatz-Wielandt max-min characterization shows one of the many one-to-one cor...

In this paper we use preferred and quasi-preferred bases of generalized eigenspaces associated with the spectral radius of nonnegative matrices to analyze the existence and uniqueness of a variant of the Jordan canonical form which we call FrobeniusJordan form. It is a combination of the classical Jordan canonical form in the part associated with the eigenvalues that are different from the spec...

in this paper some upper and lower bounds for the greatest eigenvalues of the pi and vertex pimatrices of a graph g are obtained. those graphs for which these bounds are best possible arecharacterized.

Using ergodic theory, in this paper we present a Gel’fand-type spectral radius formula which states that the joint spectral radius is equal to the generalized spectral radius for a matrix multiplicative semigroup S restricted to a subset that need not carry the algebraic structure of S. This generalizes the Berger-Wang formula. Using it as a tool, we study the absolute exponential stability of ...

let ‎x be an ‎‎n-‎‎‎‎‎‎square complex matrix with the ‎cartesian decomposition ‎‎x = a + i ‎b‎‎‎‎‎, ‎where ‎‎a ‎and ‎‎b ‎are ‎‎‎n ‎‎times n‎ ‎hermitian ‎matrices. ‎it ‎is ‎known ‎that ‎‎$vert x vert_p^2 ‎leq 2(vert a vert_p^2 + vert b vert_p^2)‎‎‎$, ‎where ‎‎$‎p ‎‎geq 2‎$‎ ‎and ‎‎$‎‎vert . vert_p$ ‎is ‎the ‎schatten ‎‎‎‎p-norm.‎ ‎‎ ‎‎in this paper‎, this inequality and some of its improvements ...

In this paper, we introduce the concept of generalized quasi-contractions in the setting of cone $b$-metric spaces over Banach algebras. By omitting the  assumption of normality we establish common fixed point theorems for the generalized quasi-contractions  with the spectral radius $r(lambda)$ of the quasi-contractive constant vector $lambda$ satisfying $r(lambda)in [0,frac{1}{s})$  in the set...