# How to calculate the rank of a tensor?

I was studying a little of tensorial calculus and came up with this problem:

Given a tensor with a rank of (0,2), $$T_{\alpha\beta}$$. Calculate the rank of this tensor $$T_{\alpha\beta}T_{\gamma}^{\sigma}T^{\beta\gamma}$$

P.D. I´m self-studying tensorial calculus, but the reference book that I have been using is not that good, so I have been trying to find the method to solve the problem but I have been unable to do so. I searched in the book first (as that is where the problem is mentioned) but there is nothing that can help me. Also, I have been trying to find an answer on the internet but the only answers are from programing... Maybe it is really simple and I cant see it, and that is why I can't find an answer. Any help would be really appreciated.

• yes @d_b , it is Rank, it is an error of translation, I'm so sorry Commented Apr 27, 2022 at 1:22
• @AFG, I double-checked and I already updated it to the correct index. What a blunder. Commented Apr 27, 2022 at 1:26

Just count unpaired indices. You have $$\alpha$$ down and $$\sigma$$ up, so you have a (1,1) tensor. (In the same way, you know that $$T_{\alpha\beta}$$ is (0,2).)
(FYI unless $$T_\gamma^\sigma$$ is symmetric you should never notate it like that, and even then I'd consider it annoying. Write $${T_\gamma}^\sigma$$ or $${T^\sigma}_\gamma,$$ since in general these could be different.)