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.

[1] http://www.ita.uni-heidelberg.de/~dullemond/lectures/tensor/tensor.pdf


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 :
$t^{\mu \nu} = v^\mu w^\nu\vec{e_{\mu}}\otimes \vec{e_{\nu}} - w^\nu v^\mu \vec{e_{\nu}}\otimes \vec{e_{\mu}}= v^\mu w^\nu\vec{e_{\mu}}\otimes \vec{e_{\nu}} - w^\mu v^\nu \vec{e_{\mu}}\otimes \vec{e_{\nu}}= \left(v^\mu w^\nu - w^\mu v^\nu \right)\vec{e_{\mu}}\otimes \vec{e_{\nu}}$

Or your example :

$ \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} $

More explanation in this wikipedia article : https://en.wikipedia.org/wiki/Cartesian_tensor

And here : Differences between a matrix and a tensor


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