## Banach Journal of Mathematical Analysis

- Banach J. Math. Anal.
- Volume 9, Number 1 (2015), 17-26.

### Some results on matrix polynomials in the max algebra

Neda Ghasemizadeh and Gholamreza Aghamollaei

#### Abstract

For any $n \times n$ nonnegative matrix $A$, and any norm $\|.\|$ on $\mathbb{R}^n$, $\eta_{\|.\|}(A)$ is defined as $ \sup\ \{\frac{\|A \otimes x\|}{\|x\|} :\ x\in \mathbb{R}_+^n \ , \ x\neq 0\}.$ Let $P(\lambda)$ be a matrix polynomial in the max algebra. In this paper, we introduce $\eta_{\|.\|}[P(\lambda)]$, as a generalization of the matrix norm $\eta_{\|.\|}(.)$, and we investigate some algebraic properties of this notion. We also study some properties of the maximum circuit geometric mean of the companion matrix of $P(\lambda)$ and the relationship between this concept and the matrices $P(1)$ and coefficients of $P(\lambda)$. Some properties of $\eta_{\|.\|}(\Psi)$, for a bounded set of max matrix polynomials $\Psi$, are also investigated.

#### Article information

**Source**

Banach J. Math. Anal., Volume 9, Number 1 (2015), 17-26.

**Dates**

First available in Project Euclid: 19 December 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.bjma/1419000574

**Digital Object Identifier**

doi:10.15352/bjma/09-1-2

**Mathematical Reviews number (MathSciNet)**

MR3296082

**Zentralblatt MATH identifier**

1312.15036

**Subjects**

Primary: 15A80: Max-plus and related algebras

Secondary: 47A30: Norms (inequalities, more than one norm, etc.) 47A56: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones) 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05] 15A18: Eigenvalues, singular values, and eigenvectors

**Keywords**

Matrix polynomials Max algebra Nonnegative matrices maximum circuit geometric mean

#### Citation

Ghasemizadeh, Neda; Aghamollaei, Gholamreza. Some results on matrix polynomials in the max algebra. Banach J. Math. Anal. 9 (2015), no. 1, 17--26. doi:10.15352/bjma/09-1-2. https://projecteuclid.org/euclid.bjma/1419000574