Normal matrices with entries from an arbitrary field of characteristic not-equal 2
Author
Pearl, Martin H.; Penn, Alan I.
SeriesTitle
Journal of Research of the National Bureau of Standards-B. Mathematical Sciences
Date
1972-07
Language
English
Keywords
Field; matrices; normality; hyperbolic space
Abstract
Let F be an arbitrary field of characteristic (=/2) not equal 2 and let conjugation in F be defined by an arbitrary involutory automorphism (possibly the identity mapping). A matrix with entries from F is normal if it commutes with its conjugate transpose. Several conditions which are equivalent to normality when F is the complex field are properly nested when F is an arbitrary field. Of these conditions, normality is the weakest and unitary diagonalizability is the strongest. When the underlying field is closed the unitarily diagonalizable matrices are those which possess a spectral representation with Hermitian idempotents. These matrices may be characterized in terms of their EP properties. A matrix is indecomposable if it has a single elementary divisor of the form (x - s)i. For i > 1, normal indecomposable matrices do not exist when F is the complex field. However there exist fields for which normal indecomposable matrices of all finite orders exist. A matrix is a normal indecomposable matrix if and only if it can be expressed as r(B ) where r(x) is a polynomial such that r'(0) =/ 0 and B is a matrix having the single elementary divisor xi and satisfying the equation B*=sB for some scalar s. When the involutory automorphism of F is the identity mapping indecomposable normal matrices of even order exist if and only if the vector space Fn is hyperbolic, and in this case the matrices are symmetric. Moreover, indecomposable norm al matrices of odd order exist if and only if Fn is the orthogonal sum of a hyperbolic space, and a one dimensional space, and in this case both symmetric and non- symmetric indecomposable normal matrices exist.
Publisher
National Bureau of Standards
Volume
76B
Issue
3-4
Page
119
Cite
J. Res. Bur. Stand., Vol. 76B, No. 3-4, p. 119
Rights
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