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)


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
    $$\rho \left( {f\left( {A_1 , \ldots ,A_m } \right)} \right) \leqslant f\left( {\rho \left( {A_1 } \right), \ldots ,\rho \left( {A_m } \right)} \right),\forall A_1 , \ldots ,A_m \in \mathbb{R}_ + ^{nn} ,\forall n \in \mathbb{N},$$
    $$\rho \left( {f\left( {A_1 , \ldots ,A_m } \right)} \right) \leqslant f\left( {\rho \left( {A_1 } \right), \ldots ,\rho \left( {A_m } \right)} \right),\forall A_1 , \ldots ,A_m \in \mathbb{R}_ + ^{nn} ,\forall n \in \mathbb{N},$$
    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.



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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Berman and R. PlemmonsNonnegative Matrices in Mathematical SciencesSIAM, Philadelphia, 1994.zbMATHCrossRefGoogle Scholar
  2. 2.
    L. Elsner and D. Hershkowitz, Hadamard functions preserving nonnegativeH-matrices, Linear Algebra Appl. 279 (1998), 13–19.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    L. Elsner, D. Hershkowitz and A. Pinkus, Functional inequalities for spectral radii of non-negative matrices, Linear Algebra Appl. 129 (1990), 103–130.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    L. Elsner, D. Hershkowitz and H. Schneider, Spectral radii of certain iteration matrices and cycle means of graphs, Linear Algebra Appl. 192 (1993), 61–81.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    L. Elsner and C.R. Johnson, Nonnegative matrices, zero patterns, and spectral inequalities, Linear Algebra Appl. 120 (1989), 225–236.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    L. Elsner, C.R. Johnson and J.A. Dias da Silva, The Perron root of a weighted geometric mean of nonnegative matrices. Linear and Multilinear Algebra 24 (1988), 1–13.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    L. Elsner and M. Neumann, Monotonic sequences and rates of convergence of asynchronized iterative methods, Linear Algebra Appl. 180 (1993), 17–33.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    G. Frobenius, Über Matrizen aus positiven ElementenS.—B. Preuss. Akad. Wiss.(1909), 471–476.Google Scholar
  9. 9.
    R.A. Horn and C.R. JohnsonMatrix AnalysisCambridge University Press, Cambridge, 1985.zbMATHGoogle Scholar
  10. 10.
    S. Karlin and F. Ost, Some monotonicity properties of Schur powers of matrices and related inequalities, Linear Algebra Appl. 68 (1985), 47–65.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • Daniel Hershkowitz
    • 1
  1. 1.Department of MathematicsTechnion — Israel Institute of TechnologyHaifaIsrael

Personalised recommendations