# On the Spectral Radius of Multi-Matrix Functions

• Daniel Hershkowitz
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 134)

## Abstract

Several problems that deal with certain spectral properties of multi-matrix functions are discussed:
1. (i)
Denote by ρ(A)the spectral radius of a nonnegative square matrix A. Known characterizations of all multi-variable functions f : ℝ + m → ℝ+ such that the Hadamard function f(A 1, …, A m ) satisfies
$$\rho \left( {f\left( {A_1 , \ldots ,A_m } \right)} \right) \leqslant f\left( {\rho \left( {A_1 } \right), \ldots ,\rho \left( {A_m } \right)} \right),\forall A_1 , \ldots ,A_m \in \mathbb{R}_ + ^{nn} ,\forall n \in \mathbb{N},$$
or
$$\rho \left( {f\left( {A_1 , \ldots ,A_m } \right)} \right) \leqslant f\left( {\rho \left( {A_1 } \right), \ldots ,\rho \left( {A_m } \right)} \right),\forall A_1 , \ldots ,A_m \in \mathbb{R}_ + ^{nn} ,\forall n \in \mathbb{N},$$
are reviewed. The study is then extended to the investigation of functions that satisfy the above conditions for just some n.

2. (ii)

For a nonnegative square matrix A denote by σ(A)the minimal real eigenvalue of its comparison matrix M(A) = 2diag(αii) – A. Denote by HP n the set of all n-by-n nonnegative H-matrices, i.e. the nonnegative matrices A for which σ(A)≥0. The relations between Hadamard functions that preserve HP n and functions that satisfy the conditions above are reviewed.

3. (iii)

Known results on the behavior of the spectral radius of products of certain one cycle matrices as a function of the lengths of the cycles are reviewed.

## Keywords

Positive Integer Linear Algebra Spectral Radius Optimal Sequence Comparison Matrix
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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