A proof of a theorem on commutative matrices
Paco A. Lagerstrom · Bulletin of the American Mathematical Society · 1945
The following theorem is well known (see, for example, Wedderburn, Lectures on matrices, p. 106) :" If the matrix B commutes with every matrix that commutes with A, then B is a scalar polynomial of A."It is thought, however, that the proof given below is simple enough to be of interest.The proof is based on the main theorem for abelian groups with a finite number of generators.The version of this theorem given in van der Waerden, Moderne Algebra, vol.2, pp.114 and 122, is especially well suited for our purpose.Let 2ft be a finite-dimensional vector space over a commutative field K. Let A be a fixed linear endomorphism of 2ft.All endomorphisms of the form P(A), where P(x) is a polynomial with coefficients in K, form a euclidean ring of operators on 2ft."Admissible subgroups" (van der Waerden, Moderne Algebra, vol. 1, p. 145) with respect to this set of operators are those subspaces of 2ft which are invariant under A. The main theorem about the decomposition of abelian groups, as applied to 2ft, then reads : There exist a finite number of subspaces 2ft» and polynomials over K, P »•(#), such that:(la) 2ft is a direct sum of the 2ft».(lb) 2ftt is invariant under A.(lc) Each 2ft» is cyclic.This means that there exist elements 0» such that each element of 2ft» is of the form P(A)ei.(Id) Piix) generates the annihilating ideal of 2ft».(le) Pi+i(x) divides P»(#).It follows that Pi(A) = 0 and that P{A) = 0 implies that Pi{x) divides P(x).(In a terminology sometimes used P,(#) is the order of e% with respect to A and P\{x) is the minimal polynomial of A. Thus the order of e\ is the minimal polynomial of A. Conversely, once the existence of an element with this property has been demonstrated, the decomposition theorem is easily proved.)We denote by Ei the projection on 2ft», that is, the linear endomorphism uniquely defined by :£»ƒ=ƒ if ƒ is in 2ft»* and £»ƒ=() if ƒ is in 2ft/, j^i.It follows that ƒ is in 2ft» if and only if £»ƒ=ƒ.An endomorphism C which commutes with Ei leaves 2ft» invariant because if ƒ is in 2ft», then EiCf = CEif= Cf.Conversely, if all 2ft» are invariant un-