# Toy example of tensor contraction

In this post I wanted to get a quick and dirty intuition of the mechanics of a tensor acting on two vectors:

Let $$\beta \in V^*$$ with coordinates in standard Cartisian basis $$\beta=\color{blue}{\begin{bmatrix}\sqrt{\pi} & \sqrt[3]{\pi} &\sqrt[5]{\pi} \end{bmatrix}}$$ and $$\gamma\in V^*$$ with coordinates $$\gamma=\color{red}{\begin{bmatrix}\frac{1}{3} &\frac{1}{5} &\frac{1}{7} \end{bmatrix}}$$. Now if we apply the tensor product $$T_{\mu\nu}\,e^\mu\otimes\,e^\nu= \beta\otimes \gamma$$ on the vectors

$$v=\color{magenta}{\begin{bmatrix}1\\7\\5\end{bmatrix}}, w = \color{orange}{\begin{bmatrix}2\\0\\3\end{bmatrix}}$$

we get

$$(\beta \otimes \gamma)[v,w]=\langle \color{blue}\beta,\color{magenta}v \rangle \times \langle \color{red}\gamma,\color{orange}w\rangle.$$

So two dot products multiplied.

$$T_{\mu\nu}{}^{\rho}\,e^\mu\otimes\,e^\nu\otimes e_\rho= \beta\otimes \gamma\otimes v ,$$

and we contracted $$\beta$$ and $$v,$$ would it be correct that we'd end up with a tensor $$T_{\nu}\,e^\nu =\langle \color{blue}\beta,\color{magenta}v \rangle \,\gamma$$

?

In other words, the contraction contributes to a scalar that comes up front as a scalar multiplying à tensor reduced in rank by $$2$$ after leaving out the two contracted elements (in this case, just a covector)?

As an amateur learner it is impossible not to revisit topics every few years. So I would not have asked this question today, and for whomever may read this, I want to point out as a NB that:\

1. Tensor contraction is properly understood as an operation within a tensor not between a tensor and another tensor (e.g. vector or covector).

2. Contracting a tensor is the equivalent of obtaining the trace of a matrix - it says something important about the matrix, but it reduces the matrix to a scalar, and hence, there is a loss of information. In the case of any other tensor, e.g. $$T^{\mu\lambda}{}_{\nu\gamma}$$, one of the contravariant indices, e.g. $$\lambda,$$ can be contracted with a covariant index such as $$\gamma,$$ provided they are of the same size by turning them into dummy indices $$T^{\mu i}{}_{\nu i}$$ and summing over (in 3 dimensional space):

$$\left(\sum_{i=1}^3 A^\lambda A_\gamma\delta^\lambda_\gamma \right) T^{\mu}{}_{\nu}$$

or

$$P^{\mu }{}_{\nu } = T^{\mu 1}{}_{\nu 1} + T^{\mu 2}{}_{\nu 2} + T^{\mu 3}{}_{\nu 3}$$

Yes, a contraction over $$\beta$$ and $$v$$ would result in $$\langle \beta, v \rangle \gamma$$: $$\delta_\rho^\mu T_{\mu\nu}{}^{\rho} = \delta_\rho^\mu \beta_\mu \gamma_\nu v^\rho = \beta_\mu \gamma_\nu v^\mu = \langle \beta, v \rangle \gamma_\nu =: T_\nu .$$
• Thank you very much! Did you make a typographic mistake in $\beta_\mu\delta_\nu v^\nu$? I would assume it was meant to be $\beta_\nu\delta_\mu v^\nu$... Commented Nov 27, 2019 at 0:21