# Product of contravariant vectors

As a CS grad student, I'm trying to learn tensors using an online resource. [1] I got stuck working through the following exercise.

The following tensor is defined of two vectors:

$$t^{\mu \upsilon} = v^\mu w^\upsilon - w^\upsilon v^\mu$$

Show that $t^{\mu \upsilon}$ is anti-symmetric.

My thought process was that $v^\mu w^\upsilon$ - $w^\upsilon v^\mu$ should be 0, because I'm essentially computing:

$$\begin{pmatrix} u^1v^1 & u^1v^2 & \cdots & u^1v^n\\ u^2v^1 & u^2v^2 & \cdots & u^2v^n\\ \vdots & \vdots & \ddots & \vdots\\ u^mv^1 & u^mv^2 & \cdots & u^mv^n\\ \end{pmatrix} - \begin{pmatrix} v^1u^1 & v^2u^1 & \cdots & v^nu^1\\ v^1u^2 & v^2u^2 & \cdots & v^nu^2\\ \vdots & \vdots & \ddots & \vdots\\ v^1u^m & v^2u^m & \cdots & v^nu^m\\ \end{pmatrix}$$

In that case, $t^{\mu \upsilon}$ would be anti-symmetric in a trivial manner. What am I missing here? As a side-note, are there any books/resources you would recommend on tensors with lots of exercises, so that it's useful for self-study? My purpose for learning is to learn enough about tensors to write software that supports tensor contractions and tensor decompositions on tensors up to rank 4.

The product of the basis vectors : $\vec{e_{\mu}}\otimes \vec{e_{\nu}}$ is not the same as $\vec{e_{\nu}}\otimes \vec{e_{\mu}}$.

While a basis for vectors is : $\vec{e_{\lambda}}$ , the basis for tensors of rank $2$ is all combinations of : $\vec{e_{\lambda}}\otimes \vec{e_{\delta}}$

So if you view the $v^{\mu}$ and $w^{\nu}$ as coordinates of vectors the product will be :
$t^{\mu \nu} = v^\mu w^\nu\vec{e_{\mu}}\otimes \vec{e_{\nu}} - w^\nu v^\mu \vec{e_{\nu}}\otimes \vec{e_{\mu}}$. So the coordinates can not be added in the way you did above to give $t^{\mu \nu} = 0$ .

The correct way to prove antisymmetry is :

$t^{\mu \nu} = v^\mu w^\nu\vec{e_{\mu}}\otimes \vec{e_{\nu}} - w^\nu v^\mu \vec{e_{\nu}}\otimes \vec{e_{\mu}} = -1 \cdot ( w^\nu v^\mu \vec{e_{\nu}}\otimes \vec{e_{\mu}} -v^\mu w^\nu\vec{e_{\mu}}\otimes \vec{e_{\nu}} ) = -1 \cdot ( w^\mu v^\nu \vec{e_{\mu}}\otimes \vec{e_{\nu}} -v^\nu w^\mu\vec{e_{\nu}}\otimes \vec{e_{\mu}} ) =-t^{\nu \mu}$

Note :
In terms of matrices of coordinates one could say :
$t^{\mu \nu} = v^\mu w^\nu\vec{e_{\mu}}\otimes \vec{e_{\nu}} - w^\nu v^\mu \vec{e_{\nu}}\otimes \vec{e_{\mu}}= v^\mu w^\nu\vec{e_{\mu}}\otimes \vec{e_{\nu}} - w^\mu v^\nu \vec{e_{\mu}}\otimes \vec{e_{\nu}}= \left(v^\mu w^\nu - w^\mu v^\nu \right)\vec{e_{\mu}}\otimes \vec{e_{\nu}}$

$\begin{pmatrix} 0 & u^1v^2-u^2v^1 & \cdots & u^1v^n-u^nv^1\\ u^2v^1-u^1v^2 & 0 & \cdots & u^2v^n-u^nv^2\\ \vdots & \vdots & \ddots & \vdots\\ u^nv^1 -u^1v^n& u^nv^2-u^2v^n & \cdots & 0\\ \end{pmatrix}$