On certain equations in matrices whose elements belong to a division algebra
Mark H. Ingraham · Bulletin of the American Mathematical Society · 1938
Introduction.A method was given by the author f to determine all matrices X having elements in a field F satisfying the matric equation PCX) =A, where P(X) is a polynomial with coefficients in F and A is a matrix with elements in F. The result gives X to within a similarity transformation commutative with A, The purely formal generalization of allowing F to be a division algebra, possibly noncommutative, not only leads to difficulties that probably can not be handled by extensions of the methods of the above mentioned paper, but seems to be devoid of interest.However, if we consider that A defines a linear transformation, we see that the answer to the following question may be of interest: Given the constantsa n , a n _i, • • • , #o, for what matrices X is ^JX^a^i^ for every vector £?If the numbers involved lie in a field, this reduces to the previously discussed problem.After defining the necessary notation, this paper proceeds to give the solution of a slightly more general problem.Consider a division algebra D and an nXn matrix A with elements in D. Let g(k) =^J\ i a i be a polynomial in X with coefficients a t -in D. If £ is an nXl matrix (vector), with elements in D, then g(A)Q% is defined^ to be^4*£a*.If gi and g 2 are the two polynomials ]£X*au an d ]CX*a 2t -, respectively, then giOg 2 =]£X''g2(X)aK.The transformation defined by %x = g(A)Oi; will be right linear if and only if the coefficients of g are in the centrum C of D where C denotes the totality of elements of D commutative with every element of D.Consider two polynomials P and Q with coefficients in D. Let A be anwXw matrix with elements in D. It is the purpose of this paper to give methods for finding all solutions X of the equation * Presented to the Society, September 8, 1937.In the preparation of this paper the author was aided by M. C. Wolf, who acted as his research assistant under appointment authorized by the Research Committee of the University of Wisconsin.t M. H. Ingraham, On the rational solutions of the matrix equation P(X)-A, Journal of Mathematics and Physics, vol.13 (1934), pp.46-50.J See M. H. Ingraham and M. C. Wolf, Relative linear sets and similarity of matrices whose elements belong to a division algebra, Transactions of this Society, vol.42 (1937), pp.16-31; referred to herein as Relative linear sets.