# Swapping indices

I have a tensor $$T^{\mu\nu}$$ that looks like this:

T^mu,nu = {{2,0,1,-1},{1,0,-3,2},{-1,1,0,0},{2,1,-1,2}}


I want to find $$T^{\nu\mu}$$.

If I swap the indices, what does that do to the matrix and for what instances am I able to say $$T^{\mu\nu} = -T^{\nu\mu}$$? I don't think that would work for this case.

Little correction: the matrix itself is $$T$$. So $$T = \begin{pmatrix} 2&0&1&-1 \\1&0&-3&2 \\ -1&1&0&0 \\ 2&1&-1&2 \end{pmatrix}$$ On the other hand, $$T^{\mu \nu}$$ is the element of the matrix at the row $$\mu$$ and column $$\nu$$ (e.g. $$T^{23} = -3$$).

Now for the question, the matrix whose elements are the same as $$T$$ but with indices swapped is the transpose of $$T$$, denoted by $$T^T$$. We swap the rows with the columns, so e.g. $$(T^T)^{32} = T^{23} = -3$$. For this matrix,

$$T^T = \begin{pmatrix} 2&1&-1&2 \\ 0&0&1&1 \\ 1&-3&0&-1 \\ -1&2&0&2 \end{pmatrix}$$ From this representation, we can see that $$T \neq - T^T$$.

• One small correction. The (2 2) part of the transposed matrix should be (2 1). Otherwise, +1. Mar 11, 2021 at 19:21
• Thank you for catching this typo! Just edited. Mar 11, 2021 at 19:23
• Sei benvenuto sempre!! Mar 11, 2021 at 19:30
• Awesome, thank you!
– Tom
Mar 14, 2021 at 20:28

You have all the components so you also have $$T^{\nu\mu}$$. You probably mean the transpose of T, but you also have that. Only for $$\mu=2$$ and $$\nu=0$$, or vice versa, you have $$T^{\mu\nu}=-T^{\nu\mu}$$.