# A tensor contraction only depends on its symmetric part if the contracted indices are symmetric in

The contraction $$T\_{a b} v^a v^b$$ computes a scalar from a tensor and a vector, and only the symmetric part of the tensor contributes. Without knowing the explicit values of  $$T\_{a b}$$ and $$v^a$$, the result represents a quadratic form whose value depends on both the tensor and the vector components.

{% embed url="<https://youtu.be/NilyMXjhkCM>" %}

{% embed url="<https://viadean.notion.site/Symmetric-and-Antisymmetric-Tensor-Contractions-2771ae7b9a32801987a7d078b8011b9f?source=copy_link>" %}