Second-Order Tensor Explained Visually | Tensor as a Linear Map

Added: 2026-06-03

[00:00:00]A second-order tensor is not just a matrix. It is a linear map from vectors to vectors. Put in a vector v, and the tensor T returns a new vector T v.

[00:00:13]The word linear is essential. If the input is alpha a plus beta b, the output is alpha T a plus beta T b. So the tensor respects vector addition and scalar multiplication.

[00:00:28]The dyadic basis explains the two indices. The operator e i tensor e j tests the input along e j, then outputs along e i. Any second-order tensor can be built as T i j e i tensor e j.

[00:00:45]Now act on a vector v equals v j e j. Using the dyadic rule, T v equals T i j v j e i. Therefore the component formula is T v sub i equals T i j v j. This is ordinary matrix multiplication written in tensor notation.

[00:01:07]Final memory map. A vector has one index. A second-order tensor has two indices. One index contracts with the input, and one index labels the output. This is why stress maps a normal vector to a traction vector.