A is defined as the matrix X satisfying the following two equations:
AXAσ2X*A,
(XA)*XA,
were σ is a given positive real number. The σ-inverse of A is denoted by Aσ. This matrix gives minimum-variance estimation in linear systems, while in contrast the Moore-Penrose inverse gives best linear unbiased estimation. The spectral properties of the σ-inverse can be represented by a complex-valued function /(zσ2). This function is continuous and bounded, which reflects the properties of the σ-inverse." />


σ-Inverse of a Matrix

Makoto SATO  Taizo IIJIMA  

Publication
IEICE TRANSACTIONS (1976-1990)   Vol.E61   No.4   pp.280-285
Publication Date: 1978/04/25
Online ISSN: 
DOI: 
Print ISSN: 0000-0000
Type of Manuscript: PAPER
Category: Data Processing
Keyword: 


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Summary: 
A new concept σ-inverse" of a matrix is proposed. The σ-inverse of a matrix A is defined as the matrix X satisfying the following two equations:
AXAσ2X*A,
(XA)*XA,
were σ is a given positive real number. The σ-inverse of A is denoted by Aσ. This matrix gives minimum-variance estimation in linear systems, while in contrast the Moore-Penrose inverse gives best linear unbiased estimation. The spectral properties of the σ-inverse can be represented by a complex-valued function /(zσ2). This function is continuous and bounded, which reflects the properties of the σ-inverse.