On the separation of eigenvalues by the permutation group

Grega Cigler; Marjan Jerman

Special Matrices (2014)

  • Volume: 2, Issue: 1, page 61-67
  • ISSN: 2300-7451

Abstract

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Let A be an invertible 3 × 3 complex matrix. It is shown that there is a 3 × 3 permutation matrix P such that the product PA has at least two distinct eigenvalues. The nilpotent complex n × n matrices A for which the products PA with all symmetric matrices P have a single spectrum are determined. It is shown that for a n × n complex matrix [...] there exists a permutation matrix P such that the product PA has at least two distinct eigenvalues.

How to cite

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Grega Cigler, and Marjan Jerman. "On the separation of eigenvalues by the permutation group." Special Matrices 2.1 (2014): 61-67. <http://eudml.org/doc/267381>.

@article{GregaCigler2014,
abstract = {Let A be an invertible 3 × 3 complex matrix. It is shown that there is a 3 × 3 permutation matrix P such that the product PA has at least two distinct eigenvalues. The nilpotent complex n × n matrices A for which the products PA with all symmetric matrices P have a single spectrum are determined. It is shown that for a n × n complex matrix [...] there exists a permutation matrix P such that the product PA has at least two distinct eigenvalues.},
author = {Grega Cigler, Marjan Jerman},
journal = {Special Matrices},
keywords = {distinct eigenvalues; irreducible group; symmetric group},
language = {eng},
number = {1},
pages = {61-67},
title = {On the separation of eigenvalues by the permutation group},
url = {http://eudml.org/doc/267381},
volume = {2},
year = {2014},
}

TY - JOUR
AU - Grega Cigler
AU - Marjan Jerman
TI - On the separation of eigenvalues by the permutation group
JO - Special Matrices
PY - 2014
VL - 2
IS - 1
SP - 61
EP - 67
AB - Let A be an invertible 3 × 3 complex matrix. It is shown that there is a 3 × 3 permutation matrix P such that the product PA has at least two distinct eigenvalues. The nilpotent complex n × n matrices A for which the products PA with all symmetric matrices P have a single spectrum are determined. It is shown that for a n × n complex matrix [...] there exists a permutation matrix P such that the product PA has at least two distinct eigenvalues.
LA - eng
KW - distinct eigenvalues; irreducible group; symmetric group
UR - http://eudml.org/doc/267381
ER -

References

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  1. [1] C. S. Ballantine, Stabilization by a diagonal matrix, Proc. Amer. Math. Soc. 25 (1970), 728-734. Zbl0228.15001
  2. [2] G. Cigler, M. Jerman, On separation of eigenvalues by certain matrix subgroups, Linear Algebra Appl. 440 (2014), 213-217.[WoS] Zbl1286.15007
  3. [3] M. Choi, Z. Huang, C. Li, N. Sze, Every invertible matrix is diagonally equivalent to a matrix with distinct eigenvalues, Linear Algebra Appl. 436 (2012), 3773-3776.[WoS] Zbl1244.15006
  4. [4] X. Feng, Z. Li, T. Huang, Is every nonsingular matrix diagonally equivalent to a matrix with all distinct eigenvalues?, Linear Algebra Appl. 436 (2012), 120-125.[WoS] Zbl1232.15008
  5. [5] M. E. Fisher, A. T. Fuller, On the stabilization of matrices and the convergence of linear iterative processes, Proc. Cambridge Philos. Soc. 54 (1958), 417-425. Zbl0085.33102
  6. [6] S. Friedland, On inverse multiplicative eigenvalue problems for matrices, Linear Algebra Appl. 12 (1975), 127-137. Zbl0329.15003
  7. [7] H. Radjavi, P. Rosenthal, Simultaneous Triangularization, Universitext, Springer-Verlag, New York, 2000. 

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