# Functions Preserving Nonnegativity of Matrices

@article{Bharali2008FunctionsPN, title={Functions Preserving Nonnegativity of Matrices}, author={Gautam Bharali and Olga Holtz}, journal={SIAM J. Matrix Anal. Appl.}, year={2008}, volume={30}, pages={84-101} }

The main goal of this work is to determine which entire functions preserve nonnegativity of matrices of a fixed order $n$—i.e., to characterize entire functions $f$ with the property that $f(A)$ is entrywise nonnegative for every entrywise nonnegative matrix $A$ of size $n\times n$. Toward this goal, we present a complete characterization of functions preserving nonnegativity of (block) upper-triangular matrices and those preserving nonnegativity of circulant matrices. We also derive necessary…

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## References

SHOWING 1-10 OF 57 REFERENCES

On computing minimal realizable spectral radii of non-negative matrices

- Mathematics, Computer ScienceNumer. Linear Algebra Appl.
- 2005

A simple bisection procedure to approximate the location of ℛ(ℒ) is proposed, which offers a quick numerical way to check whether a given n-tuple could be the spectrum of a certain non-negative matrix.

Existence and construction of nonnegative matrices with prescribed spectrum

- Mathematics
- 2003

Abstract We consider the following inverse spectrum problem for nonnegative matrices: given a set of real numbers σ={λ1,λ2,…,λn}, find necessary and sufficient conditions for the existence of an n×n…

On the comparison of some realizability criteria for the real nonnegative inverse eigenvalue problem

- Mathematics
- 2005

Abstract A result by Brauer shows how to modify one single eigenvalue of a matrix via a rank-one perturbation, without changing any of the remaining eigenvalues. This, together with the properties of…

Construction of nonnegative matrices and the inverse eigenvalue problem

- Mathematics
- 2005

This article presents a technique for combining two matrices, an n × n matrix M and an m × m matrix B, with known spectra to create an (n + m − p) × (n + m − p) matrix N whose spectrum consists of…

Row Stochastic Matrices Similar to Doubly Stochastic Matrices

- Mathematics
- 1981

The problem of determining which row stochastic n-by-n matrices are similar to doubly stochastic matrices is considered. That not all are is indicated by example, and an abstract characterization as…

The inverse eigenvalue problem for nonnegative matrices

- Mathematics
- 2004

Abstract Let A be a nonnegative matrix with spectrum ( λ 1 , λ 2 ,…, λ m ) and B be a nonnegative matrix with spectrum ( μ 1 , μ 2 ,…, μ n ), where μ 1 is the Perron eigenvalue of B . Furthermore,…

Nonnegative realization of spectra having negative real parts

- Mathematics
- 2006

Abstract The nonnegative inverse eigenvalue problem (NIEP) is the problem of determining necessary and sufficient conditions for a list of complex numbers σ to be the spectrum of a nonnegative…

Resolution of the Symmetric Nonnegative Inverse Eigenvalue Problem for Matrices Subordinate to a Bipartite Graph

- Mathematics
- 2004

There is a symmetric nonnegative matrix A, subordinate to a given bipartite graph G on n vertices, with eigenvalues λ1≥λ2≥≥λn if and only if, λ1 + λn≥0, λ2 + λn-1≥0,...,λm + λn - m + 1≥0, λm +…

The nonnegative inverse eigenvalue problem

- Mathematics
- 2004

Abstract Let σ =( λ 1 ,…, λ n ) be a list of complex numbers. The nonnegative inverse eigenvalue problem (NIEP) is the problem of determining necessary and sufficient conditions in order that σ be…

On an inverse problem for nonnegative and eventually nonnegative matrices

- Mathematics
- 1978

Let σ= (λ1,···λn)⊂C. We discuss conditions for which σ is the spectrum of a nonnegative or eventually nonnegative matrix. This brings us to study rational functions with nonnegative Maclaurin…