## 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.

### 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
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