Every invertible matrix is diagonally equivalent to a matrix with distinct eigenvalues

Man Duen Choi; Zejun Huang; Chi Kwong Li; Nung Sing Sze

doi:10.1016/j.laa.2011.12.010

Every invertible matrix is diagonally equivalent to a matrix with distinct eigenvalues

Man Duen Choi, Zejun Huang, Chi Kwong Li, Nung Sing Sze

Research output: Journal article publication › Journal article › Academic research › peer-review

8 Citations (Scopus)

Abstract

We show that for every invertible n×n complex matrix A there is an n×n diagonal invertible D such that AD has distinct eigenvalues. Using this result, we affirm a conjecture of Feng, Li, and Huang that an n×n matrix is not diagonally equivalent to a matrix with distinct eigenvalues if and only if it is singular and all its principal minors of size n-1 are zero.

Original language	English
Pages (from-to)	3773-3776
Number of pages	4
Journal	Linear Algebra and Its Applications
Volume	436
Issue number	9
DOIs	https://doi.org/10.1016/j.laa.2011.12.010
Publication status	Published - 1 May 2012

Keywords

Diagonal matrices
Distinct eigenvalues
Invertible matrices

ASJC Scopus subject areas

Algebra and Number Theory
Numerical Analysis
Geometry and Topology
Discrete Mathematics and Combinatorics

More information

10.1016/j.laa.2011.12.010

Cite this

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abstract = "We show that for every invertible n×n complex matrix A there is an n×n diagonal invertible D such that AD has distinct eigenvalues. Using this result, we affirm a conjecture of Feng, Li, and Huang that an n×n matrix is not diagonally equivalent to a matrix with distinct eigenvalues if and only if it is singular and all its principal minors of size n-1 are zero.",

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AU - Choi, Man Duen

AU - Huang, Zejun

AU - Li, Chi Kwong

AU - Sze, Nung Sing

PY - 2012/5/1

Y1 - 2012/5/1

N2 - We show that for every invertible n×n complex matrix A there is an n×n diagonal invertible D such that AD has distinct eigenvalues. Using this result, we affirm a conjecture of Feng, Li, and Huang that an n×n matrix is not diagonally equivalent to a matrix with distinct eigenvalues if and only if it is singular and all its principal minors of size n-1 are zero.

AB - We show that for every invertible n×n complex matrix A there is an n×n diagonal invertible D such that AD has distinct eigenvalues. Using this result, we affirm a conjecture of Feng, Li, and Huang that an n×n matrix is not diagonally equivalent to a matrix with distinct eigenvalues if and only if it is singular and all its principal minors of size n-1 are zero.

KW - Diagonal matrices

KW - Distinct eigenvalues

KW - Invertible matrices

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DO - 10.1016/j.laa.2011.12.010

M3 - Journal article

SN - 0024-3795

VL - 436

SP - 3773

EP - 3776

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 9

ER -

Every invertible matrix is diagonally equivalent to a matrix with distinct eigenvalues

Abstract

Keywords

ASJC Scopus subject areas

More information

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