ETD

• 2-slice tensors
• apolarity
• Kronecker form
• Macaulay2
• matrix pencils
• nonabelian apolarity
• orbits
• pencil of quadrics
• Segre classification
• tensor rank

In my work I study the Kronecker decomposition of matrix pencils and I apply it to the classification of pencils of quadrics and to the tensor rank decomposition. The Segre classification of pencils of quadrics has been studied by both in the algebraic terms of Kronecker invariants and in the geometric terms of the projective line and the base locus which each pencil defines. In the theory of tensor rank decomposition the Kronecker form allows to determine rank and border rank of tensors in $\mathbb K^2\otimes \mathbb K^m \otimes \mathbb K^n$ and $\mathbb K^2 \otimes \Sym^2(\mathbb K^m)$: I determine such ranks and the dimensions of the $\GL$-orbits. I also introduce the modern nonabelian apolarity and show how Kronecker decomposition applies. The whole work is accompanied by implementations on Macaulay2 and tables which resume examples in small dimensions.