Trace in Einstein notation I know quite well what the trace of a matrix is; however, I am not quite sure I understand the meaning of the 'trace' concept when applied to tensors. I would be very grateful to you if:
1) You could illustrate me the similarities and differences between trace in matrices and trace in tensors
2) You could illustrate me, in the easiest possible way, how traces are to be found within the Einstein notation, if at all. Does tensor contraction somehow erase the path of the trace?
Thanks in advance. 
 A: Your question indicates you should read a book on tensors or multilinear algebra, but here is a brief reply.
Tensors of type (m,n) on a vector space can be viewed as objects belonging to the space $V \otimes V \ldots \text{(m times)} V^* \otimes V^* \otimes \ldots \text{(n times)}$, where $V^*$ is the dual space to the vector space $V$ and $\otimes$ is the tensor product.
Because, by definition, dual spaces consist of one-forms that take in a vector and return a quantity in the scalar field of the vector space, there is a natural operation, called contraction, that does precisely this. So if you take an (m,n) tensor and contract one pair of indices you are left with a (m-1,n-1) type tensor. Note that the contraction inherently requires the pairing of an index representing a co-vector and an index representing a vector.
In the special case of a linear map, which is a tensor of type (1,1), the contraction operation is called a trace. It returns a scalar.
In index notation, you'd write the trace of a tensor $T^i\;_j$ as $T^i_i$.
The significance of this quantity is that it is independent of the basis you choose to find the components $T^i_j$ of the linear map T. It is an invariant function of the linear map itself. No such importance is attached to the sum of diagonal components of say a (0,2) tensor (which you might know as a quadratic form) - it would change with a change of basis.
