A Pair of Matrices with Property P
Hans Schneider · American Mathematical Monthly · 1955
A set AI,...,A. of n Xn matrices with coefficients in an algebraically closed field is said to have property P if there exists an ordering a~l), •• • , ~s), i = 1,..., n, of the characteristic roots of A l, •• •,A. for which the characteristic roots of any polynomial peAl,...,As) are p(all>,...,a~s», i = 1,...,n. In 1936 McCoy [3] proved that the set AI,..., A. has property P if and only if every matrix of the form (AiAj-AjAi)R, where R is a polynomial in the Ai, is nilpotent (for an elementary proof see Drazin, Dungey and Gruenberg [2 D. More recently two very special cases of this theorem have been proved separately. Thus in 1950 Parker [5] showed that if AB =B2=0, then the characteristic roots of A + B are the same as those of A. (The characteristic roots of B are all zero.) In 1953 Perfect [6], completing a theorem of A. Brauer [1], showed that if (C- 'AI) v = 0, and B is a matrix of rank 1 all of whose columns are multiples of the column vector v, then the characteristic roots of C+B are obtained from those of C by replacing one 'A by 'A + trace B. (One characteristic root of B is trace B, the rest are zero, andAB =0 if A = (C-'AI).) We shall give a very simple proof of a special case of McCoy's theorem which includes the two results quoted above. In order to make our theorem applicable to nXn matrices with coefficients in a field which is not necessarily algebraically closed we shall state the result in terms of the characteristic polynomial I xl-A I of a square matrix A. LEMMA. Let AI,..., A. be a set of n X n matrices with coefficients in k. If (Ltl A i)A i+l =0, for j=l,..., s-l, then the characteristic polynomial of Li-l AJ is (IIj~1 I xl-Ail)lx(8-1)".