 In theoretical physics, especially in the theory of relativity and quantum mechanics, we will regularly encounter symmetric and anti-symmetric tensors. A symmetric tensor Tij remains the same if we swap its indices. Specifically, swapping the indices of the second level tensor as a matrix means that the matrix remains the same if we transpose it, in other words, if we mirror the rows and columns on the diagonal. This symmetry property of tensors is very useful and simplifies calculations in computer physics enormously. Moreover, this property is crucial in quantum mechanics, because symmetric matrices have real eigenvalues. They therefore represent physical quantities, we call them observables, that we can measure in an experiment. So if you have a symmetric tensor in front of you, as a theoretical physicist, you should immediately get a dopamine kick. The Kronecker delta, for example, is an example of a simple symmetric tensor. We have considered a second order tensor. What if the tensor is of a higher order? What about its symmetry property then? For example, if the tensor has four indices and it remains equal when we swap the first two indices, then we are talking about a symmetric tensor in the indices m and k. However, we will also encounter tensors that are anti-symmetric. An anti-symmetric tensor changes sign when we swap its indices. If the anti-symmetric tensor is represented as a matrix, then it is equal to its negative transpose. Unfortunately, most tensors are neither symmetric nor anti-symmetric. But the great thing is, mathematically, we can split every tensor T into a symmetric and an anti-symmetric part. T is equal to s plus a. Let's take a look at how we practically split a general second level tensor T into its two parts. The symmetric part s of the tensor T is one and a half times t ij plus tji. In the second term, we have swapped the two indices. Then we added the two tensors together. The factor of one and a half is important because we have counted the tensor twice here. The anti-symmetric part a of the tensor T is one and a half times t ij minus tji. Here we have swapped the two indices, which gives the swapped tensor a minus sign. The factor of one and a half must not be missing here either. We then add the symmetric and anti-symmetric parts together to obtain the total tensor. The first summand is the symmetric part of the tensor tij. And the second summand is the anti-symmetric part.