Spaces of generators for matrix algebras with involution

Let $k$ be an algebraically closed field of characteristic different from 2. Up to isomorphism, the algebra $operatorname{Mat}_{n times n}(k)$ can be endowed with a $k$-linear involution in one way if $n$ is odd and in two ways if $n$ is even. In this paper, we consider $r$-tuples $A_bullet in operatorname{Mat}_{ntimes n}(k)^r$ such that the entries of $A_bullet$ fail to generate $operatorname{Mat}_{ntimes n}(k)$ as an algebra with involution. We show that the locus of such $r$-tuples forms a closed subvariety $Z(r;V)$ of $operatorname{Mat}_{ntimes n}(k)^r$ that is not irreducible. We describe the irreducible components and we calculate the dimension of the largest component of $Z(r;V)$ in all cases. This gives a numerical answer to the question of how generic it is for an $r$-tuple $(a_1, dots, a_r)$ of elements in $operatorname{Mat}_{ntimes n}(k)$ to generate it as an algebra with involution.