Left Associates of Monic Matrices, with an Application to Unilateral Matrix Equations
James Bell · American Journal of Mathematics · 1949
where each Mm (m 0, 1, 2, ..., k) is a matrix with elements in 5. If Milk I 0, M is said to be of degree k in A. When M74 is non-singular, M is said to be proper of degree k in A. In particular, if Mk is the identity matrix I, the matrix M will be called monic 2 of degree k. A matrix T with elements in 5 [A] is unimodular if it has an inverse whose elements are also in 5 [A]. Two matrices A and B are called left associates if there exists a unimodular matrix T such that TA = B. The relationship of left associate is an equals relationship [1]. The totality of n X n matrices with elements in 5 [A] may be divided into classes of left associates. Each class is represented by a unique matrix in canonical triangular form. This canonical triangular matrix is a matrix having all of the elements below the main diagonal equal to zero. The diagonal elements if they are not zero, are monic polynomials. The elements of each column are reduced modulo the main diagonal element when that element is not zero. If the main diagonal element is zero, then every element of the row, in which it occurs, is zero.3 The problem considered is that of determining under what conditions a matrix will be the left associate of a monic matrix of degree k. The special case of this problem, arising when k1, is of value in the completion of