# How does contraction apply to rank-1 tensors?

I'm just learning about tensors and the tensor product, and have been given examples for tensors such as $$a \equiv x_1^2+x_2^2$$ being one because it is the contraction of the tensorproduct of $$\{x_i\}$$ with itself, $$\{x_i\}$$ being a rank-1 tensor in 2 dimensions. I have only encountered the contraction as the trace of matrix, so how does it apply to this case? Also, can I understand $$\{x_i\}$$ as a $$2 \times 2$$ matrix and for example show that a new variable $$b \equiv x_1+x_2$$ is a tensor because it is the trace of the tensor $$\{x_i\}$$?

• A rank one tensor is of type $(1,0)$ or of type $(0,1)$. As the contraction of a $(p,q)$ tensor gives birth to a $(p-1,q-1)$ tensor, here it would have no sense because it would have a covariance or contravariance $-1$ – DIdier_ May 27 at 13:26