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Eigenconfigurations of tensors

Square matrices A {\displaystyle A} with entries in a field K {\displaystyle K} represent linear maps of vector spaces, say K n → K n {\displaystyle K^{n}\to K^{n}}, and thus linear maps ψ: P n − 1 → P n − 1 {\displaystyle \psi :\mathbb {P} ^{n-1}\to \mathbb {P} ^{n-1}} of projective spaces over K {\displaystyle K}. If A {\displaystyle A} is nonsingular then ψ {\displaystyle \psi } is well-defined everywhere, and the eigenvectors of A {\displaystyle A} correspond to the fixed points of ψ {\displaystyle \psi }. The eigenconfiguration of A {\displaystyle A} consists of n {\displaystyle n} points in P n − 1 {\displaystyle \mathbb {P} ^{n-1}}, provided A {\displaystyle A} is generic and K {\displaystyle K} is algebraically closed. The fixed points of nonlinear maps are the eigenvectors of tensors. Let A = (a i 1 i 2 ⋯ i d) {\displaystyle A=(a_{i_{1}i_{2}\cdots i_{d}})} be a d {\displaystyle d} -dimensional tensor of format n × n × ⋯ × n {\displaystyle n\times n\times \cdots \times n} with entries (a i 1 i 2 ⋯ i d) {\displaystyle (a_{i_{1}i_{2}\cdots i_{d}})} lying in an algebraically closed field K {\displaystyle K} of characteristic zero. Such a tensor A ∈ (K n) ⊗ d {\displaystyle A\in (K^{n})^{\otimes d}} defines polynomial maps K n → K n {\displaystyle K^{n}\to K^{n}} and P n − 1 → P n − 1 {\displaystyle \mathbb {P} ^{n-1}\to \mathbb {P} ^{n-1}} with coordinates: ψ i (x 1, …, x n) = ∑ j 2 = 1 n ∑ j 3 = 1 n ⋯ ∑ j d = 1 n a i j 2 j 3 ⋯ j d x j 2 x j 3 ⋯ x j d for i = 1, …, n {\displaystyle \psi _{i}(x_{1},\ldots,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,\ldots,n}

Thus each of the n {\displaystyle n} coordinates of ψ {\displaystyle \psi } is a homogeneous polynomial ψ i {\displaystyle \psi _{i}} of degree d − 1 {\displaystyle d-1} in x = (x 1, …, x n) {\displaystyle \mathbf {x} =\left(x_{1},\ldots,x_{n}\right)}. The eigenvectors of A {\displaystyle A} are the solutions of the constraint: rank (x 1 x 2 ⋯ x n ψ 1 (x) ψ 2 (x) ⋯ ψ n (x)) ≤ 1 {\displaystyle {\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 × 2 {\displaystyle 2\times 2} minors of this matrix.