The Algebra Generated by Three Commuting Matrices

We present a survey of an open problem concerning the dimension of the algebra generated by three commuting matrices. This article concerns a problem in algebra that is completely elementary to state, yet, has proven tantalizingly difficult and is as yet unsolved. Consider C[A,B,C] , the C-subalgebra of the n × n matrices Mn(C) generated by three commuting matrices A, B, and C. Thus, C[A,B,C] consists of all Clinear combinations of “monomials” AiBjCk, where i, j, and k range from 0 to infinity. Note that C[A,B,C] and Mn(C) are naturally vector-spaces over C; moreover, C[A,B,C] is a subspace of Mn(C). The problem, quite simply, is this: Is the dimension of C[A,B,C] as a C vector space bounded above by n? Note that the dimension of C[A,B,C] is at most n2, because the dimension of Mn(C) is n2. Asking for the dimension of C[A,B,C] to be bounded above by n when A, B, and C commute is to put considerable restrictions on C[A,B,C]: this is to require that C[A,B,C] occupy only a small portion of the ambient Mn(C) space in which it sits. Actually, the dimension of C[A,B,C] is already bounded above by something slightly smaller than n2, thanks to a classical theorem of Schur ([16]), who showed that the maximum possible dimension of a commutative Csubalgebra of Mn(C) is 1 + bn2/4c. But n is small relative even to this number. To understand the interest in n being an upper bound for the dimension of C[A,B,C], let us look more generally at the dimension of the C-subalgebra of Mn(C) generated by k-commuting matrices. Let us start with the k = 1 case: note that “one commuting matrix” is just an arbitrary matrix A. Recall that the Cayley-Hamilton theorem tells us that An is a linear combination of I, A, . . . , An−1, where I stands for the identity matrix. From this, it follows by repeated reduction that An+1, An+2, etc. are all linear combinations of I, A, . . . , An−1 Thus, C[A], the C-subalgebra of Mn(C) generated by A, is of dimension at most n, and this is just a simple consequence of CayleyHamilton. The case k = 2 is therefore the first significant case. It was treated by Gerstenhaber ([4]) as well as Motzkin and Taussky-Todd ([13], who proved independently that the variety of commuting pairs of matrices is irreducible. It follows from this that if A and B are two commuting matrices, then too,