Upper and lower bounds for ranks of matrix expressions using generalized inverses

The maximal and minimal ranks of the matrix expression A 1− B 1 X 1 C 1− B 2 X 2 C 2 with respect to X 1 and X 2 are presented. As applications, the maximal and minimal ranks of A 1− B 1 XC 1 subject to a consistent matrix equation B 2 XC 2= A 2 are also determined. In addition, the maximal and mini...

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Bibliographic Details
Published inLinear algebra and its applications Vol. 355; no. 1; pp. 187 - 214
Main Author Tian, Yongge
Format Journal Article
LanguageEnglish
Published New York, NY Elsevier Inc 01.11.2002
Elsevier Science
Subjects
Algebra
Exact sciences and technology
Generalized inverse
Invariance
Linear and multilinear algebra, matrix theory
Mathematics
Matrix equation
Matrix expression
Range
Rank
Schur complement
Sciences and techniques of general use
15A24
Matrix expression
Rank
15A03
Invariance
Range
Schur complement
15A09
Generalized inverse
Matrix equation
Lower bound
Upper bound
Operator equation
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ISSN0024-3795
1873-1856
DOI10.1016/S0024-3795(02)00345-2

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Summary:The maximal and minimal ranks of the matrix expression A 1− B 1 X 1 C 1− B 2 X 2 C 2 with respect to X 1 and X 2 are presented. As applications, the maximal and minimal ranks of A 1− B 1 XC 1 subject to a consistent matrix equation B 2 XC 2= A 2 are also determined. In addition, the maximal and minimal ranks of the Schur complement D− CA − B with respect to the generalized inverse A − of A and their various consequences are also considered.
ISSN:0024-3795
1873-1856
DOI:10.1016/S0024-3795(02)00345-2