Generalized equivalence of pairs of matrices
V. М. Petrychkovych · Linear and Multilinear Algebra · 2000
Pairs (A 1 B 1) and (A 2 B 2) of matrices over a principal ideal domain R are called the generalized equivalent pairs if A 2=UA 1 V 1 B 2=UB 1 V 2 for some invertible matrices U V 1 V 2 over R. A special form is established to which a pair of matrices can be reduced by means of generalized equivalent transformations. Besides necessary and sufficient conditions are found, under which a pair of matrices is generalized equivalent to a pair of diagonal matrices. Applications are made to study the divisibility of matrices and multiplicative property of the Smith normal form.