Editing Tensor product (section)

== Eigenconfigurations of tensors ==
Square [[Matrix (mathematics)|matrices]] <math>A</math> with entries in a [[Field (mathematics)|field]] <math>K</math> represent [[linear map]]s of [[vector space]]s, say {{tmath|1= K^n \to K^n }}, and thus linear maps <math>\psi : \mathbb{P}^{n-1} \to \mathbb{P}^{n-1}</math> of [[projective spaces]] over {{tmath|1= K }}. If <math>A</math> is [[Invertible matrix|nonsingular]] then <math>\psi</math> is [[well-defined]] everywhere, and the [[Eigenvalues and eigenvectors|eigenvectors]] of <math>A</math> correspond to the fixed points of {{tmath|1= \psi }}. The ''eigenconfiguration'' of <math>A</math> consists of <math>n</math> points in {{tmath|1= \mathbb{P}^{n-1} }}, provided <math>A</math> is generic and <math>K</math> is [[Algebraically closed field|algebraically closed]]. The fixed points of nonlinear maps are the eigenvectors of tensors. Let <math>A = (a_{i_1 i_2 \cdots i_d})</math> be a <math>d</math>-dimensional tensor of format <math>n \times n \times \cdots \times n</math> with entries <math>(a_{i_1 i_2 \cdots i_d})</math> lying in an algebraically closed field <math>K</math> of [[Characteristic (algebra)|characteristic]] zero. Such a tensor <math>A \in (K^{n})^{\otimes d}</math> defines [[Morphism of algebraic varieties|polynomial maps]] <math>K^n \to K^n</math> and <math>\mathbb{P}^{n-1} \to \mathbb{P}^{n-1}</math> with coordinates:
<math display="block">\psi_i(x_1, \ldots, 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, \ldots, n</math>

Thus each of the <math>n</math> coordinates of <math>\psi</math> is a [[homogeneous polynomial]] <math>\psi_i</math> of degree <math>d-1</math> in {{tmath|1= \mathbf{x} = \left(x_1, \ldots, x_n\right) }}. The eigenvectors of <math>A</math> are the solutions of the constraint:
<math display="block">\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
</math>
and the eigenconfiguration is given by the [[Algebraic variety|variety]] of the <math>2 \times 2</math> [[Minor (linear algebra)|minors]] of this matrix.<ref>{{cite arXiv |last1=Abo |first1=H. |last2=Seigal |first2=A. |author2-link=Anna Seigal |last3=Sturmfels |first3=B. |author3-link=Bernd Sturmfels |title=Eigenconfigurations of Tensors |date=2015 |class=math.AG |eprint=1505.05729 }}</ref>