Reference: Moore, Thomas A., *A General Relativity Workbook*, University Science Books (2013) – Chapter 6; Problem 6.6.

Tensors, like matrices, can be symmetric or anti-symmetric. Since a tensor can have a rank higher than 2, however, a single tensor can have more than one symmetry. For a rank-2 tensor , it is symmetric if and anti-symmetric if . In matrix terminology, a symmetric rank-2 tensor is equal to its transpose, and an anti-symmetric rank-2 tensor is equal to the negative of its transpose.

A higher rank tensor can be symmetric or anti-symmetric in any pair of its indices, provided both indices are either upper or lower. For example, can be symmetric or anti-symmetric in any pair selected from or in the pair , but not in one upper and one lower index. Thus if and , then is symmetric in and , and anti-symmetric in and .

Returning to rank-2 tensors, we can show that the symmetry property is an invariant:

If a tensor is symmetric in a pair of upper indices, then if both indices are lowered, the resulting tensor is also symmetric in the two lower indices:

Similarly, the anti-symmetric property persists through lowering of indices. If

If then all diagonal elements must be zero, since has only zero as a solution. Also, the trace is

In line 3, we used , since in terms of the basis vectors, , and thus the metric tensor is symmetric. Thus the trace is also zero for an anti-symmetric tensor.

A rank 2 symmetric tensor in dimensions has all the diagonal elements and the upper (or lower) triangular set of elements as independent components, so the total number of independent elements is . An anti-symmetric tensor has zeroes on the diagonal, so it has independent elements.

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SerkanI think something wrong about last two summation. If there is n element in matrix, that matrix has n number of elements not n(n+1)/2. It might be about something about tensor; I would like to know.

growescienceAn matrix has elements, but if the matrix is symmetric, then all elements above the diagonal are the same as those below the diagonal. Thus the number of independent elements is 1 in the first row, 2 in the second row and so on, up to in the th row. The total number of independent elements is thus as stated. I’ve just used the standard formula for the sum of integers from 1 to , which is .

SerkanI thought matrix likes

1 2 3 4

5 6 7 8

……. n

but what you are saying is

independent elements

1 1

1 1 2

1 1 1 3

1 1 1 1 4

1 1 1 . ……. 1 n