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**An equivalence canonical form of a matrix triplet over an arbitrary division ring with applications.**
*(English)*
Zbl 1218.15008

The authors give a decomposition concerning the general matrix triplet over an arbitrary division ring \(\mathcal{F}\) with the same row or column numbers. They also design a practical algorithm for the decomposition of the matrix triplet. As applications, they present necessary and sufficient conditions for the existence of the general solutions to the system of matrix equations

\[ DXA = C_1 , \quad EXB = C_2 , \quad FXC = C_3 \]

and the matrix equation

\[ AXD + BYE + CZF = G \]

over \(\mathcal{F}\). They give the expressions of the general solutions to the system and the matrix equation when the solvability conditions are satisfied. Moreover, they present numerical examples to illustrate the results of this paper. They also mention the other applications of the equivalence canonical form, for instance, for the compression of color images.

\[ DXA = C_1 , \quad EXB = C_2 , \quad FXC = C_3 \]

and the matrix equation

\[ AXD + BYE + CZF = G \]

over \(\mathcal{F}\). They give the expressions of the general solutions to the system and the matrix equation when the solvability conditions are satisfied. Moreover, they present numerical examples to illustrate the results of this paper. They also mention the other applications of the equivalence canonical form, for instance, for the compression of color images.

Reviewer: A. Arvanitoyeorgos (Patras)

### MSC:

15A21 | Canonical forms, reductions, classification |

15A22 | Matrix pencils |

15A24 | Matrix equations and identities |

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

15A03 | Vector spaces, linear dependence, rank, lineability |

16K20 | Finite-dimensional division rings |

65F30 | Other matrix algorithms (MSC2010) |

94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |

### Keywords:

division ring; linear matrix equation; equivalence canonical form of a matrix triplet; algorithm; system of matrix equations; numerical examples; compression of color images
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\textit{Q. Wang} et al., Sci. China, Math. 54, No. 5, 907--924 (2011; Zbl 1218.15008)

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