Diagonal Similarity and Equivalence for Matrices

the cyclic products of matrices and diagonal similarity. In this paper we consider diagonal similarity for matrices, which may be infinite, and whose elements lie in a (possible non-commutative) group G with O. Let H be a subgroup of a group G and let A be an irreducible square matrix with entries in GO. In Theorem 3.4, we give necessary and sufficient conditions for the existence of a matrix B with entries in H O which is diagonally similar to A. If H is a complete lattice ordered group whose positive cone H+ is normal in G, we give necessary and sufficient conditions for the existence of a matrix ' B in (H+)O which is diagonally similar to A; see Theorems 4.1 and 4.2. Our Theorem 4.1 reduces to a result due to AFRIAT [1J, [2J Theorem 2 and FIEDLER-PTAK [4J Theorem 2.2 in the case when G and H are the additive group of reals, there is no absorbing (zero) element and A is a finite matrix. Let A be a rectangular matrix, possible infinite, with entries in GO, such that each row and column has at least one element in G. We construct a square matrix Alp of larger siz::, which is always completely reducible, Corollary 5.4. In Theorem 5.6, we show that two recta:ngular matrices A and B are diagonally equivalent if and only if Alp and Blp are diagonally similar. Thus it is possible to derive theorems on diagonal equivalence for arbitrary rectangular matrices from theorems on the diagonal similarity of irreducible square matrices, e.g. Theorem 6.3. In particular, as a corollary to either Theorem 6.2 or Theorem 6.3 we obtain a slightly improved version of the remarkable result by LALLEMENT-PETRICH ([6J, Theorem 1 (b)-= (e),