Rational methods in matrix equations
Mark H. Ingraham · Bulletin of the American Mathematical Society · 1941
I. Introduction. 2 I shall start with what I hope will not prove an overelaborate statement of the limitations of this paper in scope and treatment.I shall assume throughout a knowledge of the definitions of a field, a division algebra, of matrices and rational operations thereon.I shall also need to assume a knowledge of what is meant by the invariant factors and elementary divisors of a matrix.Little essential will be lost if the only field considered by the listeners is the rational number system and the only division algebra that of quaternions over the rational field.Consider a field $ and a system of constant matrices A i, • • • , A h unknown matrices Xi, • • • , X n and equations (1) 4>i(A u • • • ,A h Xi, • • • J B ) = 0 where the <£/s are polynomials with coefficients in $.If the elements of Ai are a%jh and of Xi are xuk, (1) is equivalent to a system (2) *.(•", *</*.• ' • ) = 0