Unitary equivalence of a matrix to its transpose

Stephan Ramon Garcia; James E. Tener

Unitary equivalence of a matrix to its transpose

Stephan Ramon Garcia^*, James E. Tener

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

29 Citations (Scopus)

Abstract

Motivated by a problem of Halmos, we obtain a canonical decomposition for complex matrices which are unitarily equivalent to their transpose (UET). Surprisingly, the naïve assertion that a matrix is UET if and only if it is unitarily equivalent to a complex symmetric matrix holds for matrices 7 × 7 and smaller, but fails for matrices 8 × 8 and larger.

Original language	English
Pages (from-to)	179-203
Number of pages	25
Journal	Journal of Operator Theory
Volume	68
Issue number	1
Publication status	Published - 2012
Externally published	Yes

Cite this

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title = "Unitary equivalence of a matrix to its transpose",

abstract = "Motivated by a problem of Halmos, we obtain a canonical decomposition for complex matrices which are unitarily equivalent to their transpose (UET). Surprisingly, the na{\"i}ve assertion that a matrix is UET if and only if it is unitarily equivalent to a complex symmetric matrix holds for matrices 7 × 7 and smaller, but fails for matrices 8 × 8 and larger.",

keywords = "Complex symmetric matrix, Complex symmetric operator, Linear preserver, Numerical range, Skew-Hamiltonian, Transpose, UECSM, Unitary equivalence, Unitary orbit",

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year = "2012",

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T1 - Unitary equivalence of a matrix to its transpose

AU - Garcia, Stephan Ramon

AU - Tener, James E.

PY - 2012

Y1 - 2012

N2 - Motivated by a problem of Halmos, we obtain a canonical decomposition for complex matrices which are unitarily equivalent to their transpose (UET). Surprisingly, the naïve assertion that a matrix is UET if and only if it is unitarily equivalent to a complex symmetric matrix holds for matrices 7 × 7 and smaller, but fails for matrices 8 × 8 and larger.

AB - Motivated by a problem of Halmos, we obtain a canonical decomposition for complex matrices which are unitarily equivalent to their transpose (UET). Surprisingly, the naïve assertion that a matrix is UET if and only if it is unitarily equivalent to a complex symmetric matrix holds for matrices 7 × 7 and smaller, but fails for matrices 8 × 8 and larger.

KW - Complex symmetric matrix

KW - Complex symmetric operator

KW - Linear preserver

KW - Numerical range

KW - Skew-Hamiltonian

KW - Transpose

KW - UECSM

KW - Unitary equivalence

KW - Unitary orbit

UR - https://www.scopus.com/pages/publications/84871364408

M3 - Article

AN - SCOPUS:84871364408

SN - 0379-4024

VL - 68

SP - 179

EP - 203

JO - Journal of Operator Theory

JF - Journal of Operator Theory

IS - 1

ER -

Unitary equivalence of a matrix to its transpose

Abstract

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