##
**Pseudo-similarity and partial unit regularity.**
*(English)*
Zbl 0544.15005

Let R be a ring and let A and B be square matrices of orders m and n respectively. A is said to be pseudo-similar to B if there is an \(m\times n\) matrix X and two possibly distinct \(n\times m\) matrices Y and Z such that \(YAX=B\), \(XAZ=A\), \(XYX=X\), \(XZX=X\). Recall that an element a in R is regular if there is a solution to the equation \(axa=a\) and unit regular (assume R has a unity) if there is a unit such that \(aua=a\). Finally R is called partially unit regular if every regular element is unit regular. The definition of pseudo-similarity can be stated for R as well.

The authors show that R is partially unit regular if and only if pseudo- similar elements of R are in fact similar. Applications are made to matrices over a principal ideal domain and to the reduction of regular matrices to a normal form.

The authors show that R is partially unit regular if and only if pseudo- similar elements of R are in fact similar. Applications are made to matrices over a principal ideal domain and to the reduction of regular matrices to a normal form.

Reviewer: G.P.Barker

### MSC:

15B33 | Matrices over special rings (quaternions, finite fields, etc.) |

15A21 | Canonical forms, reductions, classification |

15A09 | Theory of matrix inversion and generalized inverses |

PDFBibTeX
XMLCite

\textit{F. J. Hall} et al., Czech. Math. J. 33(108), 361--372 (1983; Zbl 0544.15005)

Full Text:
EuDML

### References:

[1] | T. P. Donovan: Certain matrix congruences mod \(p^n\). Ann. Mat. Pura Appl. IV, 65 (1977), 193-214. · Zbl 0374.15005 |

[2] | F. J. Hall R. E. Hartwig I. J. Katz: A note on pseudo-similarity of matrices. J. of Industrial Math. Soc. 28 (1978), 25-36. · Zbl 0398.15009 |

[3] | F. J. Hall I. J. Katz: On ranks of integral generalized inverses of integral matrices. Lin. and Mult. Alg. 7 (1979), 73-85. · Zbl 0398.15003 |

[4] | R. E. Hartwig: How to partially order regular elements. Math. Japonica 25 (1980), 1-13. · Zbl 0442.06006 |

[5] | R. E. Hartwig: Drazin inverses and canonical forms in \(M_n (Z/h)\). Lin. Alg. and Appl. 37 (1981), 205-233. · Zbl 0455.15003 |

[6] | R. E. Hartwig F. J. Hall: Pseudo-similarity for matrices over a field. Proc. of AMS, 71 (1978), 6-10. · Zbl 0386.15013 |

[7] | R. E. Hartwig J. Luh: A note on the group structure of unit regular ring elements. Pacific J. of Math. 71 (1977), 449-461. · Zbl 0355.16005 |

[8] | R. E. Hartwig M. S. Putcha: Semi-similarity for matrices over a divison ring. Lin. Alg. and Appl. 39 (1981), 125-132. · Zbl 0467.15007 |

[9] | M. Henriksen: On a class of regular rings that are elementary divisor rings. Archiv der Mathematik. 24 (1973), 133-141. · Zbl 0257.16015 |

[10] | M. Newman: Integral matrices. Academic Press, New York) · Zbl 0254.15009 |

[11] | E. D. Sontag: On generalized inverses of polynomial and other matrices. IEEE Trans. Automat. Contr. 25 (1980), 514-517. · Zbl 0447.15003 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.