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

$\psi _{i}(x_{1},...,x_{n})=\sum _{j_{2}=1}^{n}\sum _{j_{3}=1}^{n}\cdots \sum _{j_{d}=1}^{n}a_{ij_{2}j_{3}\cdots j_{d}}x_{j_{2}}x_{j_{3}}\cdots x_{j_{d}}\;\;{\mbox{for }}i=1,...,n$

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.