User:Dlmke/sandbox

Eigenconfigurations of Tensors
Square matrices $A$ with entries in a field $K$ represent linear maps of vector spaces, say $$K^n \to K^n$$, and thus linear maps $$\psi : \mathbb{P}^{n-1} \to \mathbb{P}^{n-1}$$ of projective spaces over $$K$$. If $$A$$ is nonsingular then $$\psi$$ is well-defined everywhere, and the eigenvectors of $$A$$ correspond to the fixed points of $$\psi$$. The eigenconfiguration of $$A$$ consists of $$n$$ points in $$\mathbb{P}^{n-1}$$, provided $$A$$ is generic and $$K$$ is algebraically closed. The fixed points of nonlinear maps are the eigenvectors of tensors. Let $$A = (a_{i_1 i_2 \cdots i_d})$$ be a $$d$$-dimensional tensor of format $$n \times n \times \cdots \times n$$ with entries $$(a_{i_1 i_2 \cdots i_d})$$ lying in an algebraically closed field $$K$$ of characteristic zero. Such a tensor $$A \in (K^{n})^{\otimes d}$$ defines polynomial maps $$K^n \to K^n$$ and $$\mathbb{P}^{n-1} \to \mathbb{P}^{n-1}$$ with coordinates
 * $$\psi_i(x_1, ..., x_n) = \sum_{j_2=1}^n \sum_{j_3=1}^n \cdots \sum_{j_d = 1}^n a_{i j_2 j_3 \cdots j_d} x_{j_2} x_{j_3}\cdots x_{j_d} \;\; \mbox{for } i = 1,...,n $$

Thus each of the $$n$$ coordinates of $$ \psi $$ is a homogeneous polynomial $$ \psi_i $$ of degree $$ d - 1$$ in $$\mathbf{x} = (x_1, ..., x_n)$$. The eigenvectors of $$A$$ are the solutions of the constraint
 * $$\mbox{rank} \begin{pmatrix}x_1 & x_2 & \cdots & x_n \\ \psi_1(\mathbf{x}) & \psi_2(\mathbf{x}) & \cdots & \psi_n(\mathbf{x}) \end{pmatrix} \leq 1 $$

and the eigenconfiguration is given by the variety of the $$2 \times 2$$ minors of this matrix.