A singular k-tuple of a tensor T of format (n1,…,nk) is essentially a complex critical point of the distance function from T constrained to the cone of tensors of format (n1,…,nk) of rank at most one. A generic tensor has finitely many complex singular k-tuples, and their number depends only on the tensor format. Furthermore, if we fix the first k−1 dimensions ni, then the number of singular k-tuples of a generic tensor becomes a monotone non-decreasing function in one integer variable nk, that stabilizes when (n1,…,nk) reaches a boundary format. In this paper, we study the linear span of singular k-tuples of a generic tensor. Its dimension also depends only on the tensor format. In particular, we concentrate on special order three tensors and order-k tensors of format (2,…,2,n). As a consequence, if again we fix the first k−1 dimensions ni and let nk increase, we show that in these special formats, the dimension of the linear span stabilizes as well, but at some concise non-sub-boundary format. We conjecture that this phenomenon holds for an arbitrary format with k>3. Finally, we provide equations for the linear span of singular triples of a generic order three tensor T of some special non-sub-boundary format. From these equations, we conclude that T belongs to the linear span of its singular triples, and we conjecture that this is the case for every tensor format.

Kalman varieties of tensors are algebraic varieties consisting of tensors whose singular vector k-tuples lay on prescribed subvarieties. They were first studied by Ottaviani and Sturmfels in the context of matrices. We extend recent results of Ottaviani and the first author to the partially symmetric setting. We describe a generating function whose coefficients are the degrees of these varieties and we analyze its asymptotics, providing analytic results à la Zeilberger and Pantone. We emphasize the special role of isotropic vectors in the spectral theory of tensors and describe the totally isotropic Kalman variety as a dual variety.