On Dominance and Varieties of Commuting Matrices

4), a question first raised by Goto. In the same section it is shown that the algebra generated by a pair of commuting n x n matrices can not have dimension greater than n, and that this algebra can always be embedded in a commutative algebra of dimension exactly n, a result derived by introducing the notion of a specialization of an algebra of matrices. The remainder of Chapter II contains related results of independent interest included mainly for later reference. The present paper was originally intended to be the first part of On nilalgebras and linear varieties of nilpotent matrices, IV (in preparation), which study continues the program of determining the structure of linear varieties of nilpotent matrices (nilvarieties). It is there shown that if V is a nilvariety of n x n matrices, and if c is the dimension of the algebra of all n x n matrices commuting with a generic element of V, then dim V_? 1/2 (n2 c). All nilvarieties V for which dim V l/2 (X2-c) are determined and shown to be associative nilpotent algebras of a type which we have elected to call anti-semisimple. These may be described