The weave product of two conics

TL;DR

This paper provides a geometric interpretation of the tensor diagram product of two matrices as related to common tangents of conics in projective geometry, linking algebraic tensor operations with geometric properties.

Contribution

It introduces a novel geometric perspective on the tensor diagram product of two conics, connecting algebraic matrix operations with geometric configurations in projective space.

Findings

Tensor diagram product relates to common tangents of conics

Geometric interpretation bridges algebra and projective geometry

Enhances understanding of tensor diagrams in geometric contexts

Abstract

Tensor diagrams are a handy way to depict complicated relationships between objects in projective geometry. One of the simpler ones takes two copies of a $3\times 3$ matrix and computes its adjugate. In this paper, we give a geometric interpretation of this construction when two different matrices are used. To do so, we interpret them as coefficient matrices of conic sections in the projective plane and relate the diagram construction to their common tangents.