[ Disclaimer: I am only starting to get my head around concepts of tensor calculus, so I apologize in advance for the lack of clarity or asking something trivial. I came across it trying to apply it somewhere]
I understand that a tensor can be seen a multilinear map. Is there a formal study of the kernel(s) of this map, i.e. a null-space of the tensor? I am guessing such a definition would have to take into account the tensor-rank of the tensor $T$ and the tensor-rank of tensor $X$ such that $X_IT^I = \mathbf{0} $, so we would have multiple null-spaces for one tensor $T$ each with a different tensor-rank.
If so could you provide with a reference? Is there a way to compute it as well?
A case:
In a finite dimensional vector space $V$ over the reals, if $Q:V\times V\to\mathbb R$ is a bilinear map then this could be set by $(v,w)\mapsto v^{\top}Qw$ where $Q$ is a square matrix.
If this matrix has eigenvectors $u_0$ with eigenvalue equal to zero then $v^{\top}Qu_0=0$ and if $Q^{\top}$ has also $u_1$ as an eigenvector of eigenvalue equal to zero then $(u_1^{\top}Qw)^{\top}=w^{\top}Q^{\top}u_1=0$.