# Equalities for orthogonal projectors and their operations

Open Mathematics (2010)

• Volume: 8, Issue: 5, page 855-870
• ISSN: 2391-5455

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## Abstract

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A complex square matrix A is called an orthogonal projector if A 2 = A = A*, where A* denotes the conjugate transpose of A. In this paper, we give a comprehensive investigation to matrix expressions consisting of orthogonal projectors and their properties through ranks of matrices. We first collect some well-known rank formulas for orthogonal projectors and their operations, and then establish various new rank formulas for matrix expressions composed by orthogonal projectors. As applications, we derive necessary and sufficient conditions for various equalities for orthogonal projectors and their operations to hold.

## How to cite

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Yongge Tian. "Equalities for orthogonal projectors and their operations." Open Mathematics 8.5 (2010): 855-870. <http://eudml.org/doc/269712>.

@article{YonggeTian2010,
abstract = {A complex square matrix A is called an orthogonal projector if A 2 = A = A*, where A* denotes the conjugate transpose of A. In this paper, we give a comprehensive investigation to matrix expressions consisting of orthogonal projectors and their properties through ranks of matrices. We first collect some well-known rank formulas for orthogonal projectors and their operations, and then establish various new rank formulas for matrix expressions composed by orthogonal projectors. As applications, we derive necessary and sufficient conditions for various equalities for orthogonal projectors and their operations to hold.},
author = {Yongge Tian},
journal = {Open Mathematics},
keywords = {Orthogonal projector; Idempotent matrix; Matrix equality; Rank equality; Range equality; Commutativity; Moore-Penrose inverse; Group inverse; Reverse-order law; orthogonal projector; idempotent matrix; matrix equality; rank equality; range equality; commutativity; group inverse; reverse-order law},
language = {eng},
number = {5},
pages = {855-870},
title = {Equalities for orthogonal projectors and their operations},
url = {http://eudml.org/doc/269712},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Yongge Tian
TI - Equalities for orthogonal projectors and their operations
JO - Open Mathematics
PY - 2010
VL - 8
IS - 5
SP - 855
EP - 870
AB - A complex square matrix A is called an orthogonal projector if A 2 = A = A*, where A* denotes the conjugate transpose of A. In this paper, we give a comprehensive investigation to matrix expressions consisting of orthogonal projectors and their properties through ranks of matrices. We first collect some well-known rank formulas for orthogonal projectors and their operations, and then establish various new rank formulas for matrix expressions composed by orthogonal projectors. As applications, we derive necessary and sufficient conditions for various equalities for orthogonal projectors and their operations to hold.
LA - eng
KW - Orthogonal projector; Idempotent matrix; Matrix equality; Rank equality; Range equality; Commutativity; Moore-Penrose inverse; Group inverse; Reverse-order law; orthogonal projector; idempotent matrix; matrix equality; rank equality; range equality; commutativity; group inverse; reverse-order law
UR - http://eudml.org/doc/269712
ER -

## References

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