Is it correct to represent a tensor with 2 covariant indices as a matrix of rows and columns on a paper?

Question

We represent a tensor with 1 covariant index i & 1 contravariant index j as a matrix of rows and columns on a paper. The no of rows will be j. The no of columns will be i.

Is it correct to represent a tensor with 2 covariant indices as a matrix of rows and columns on a paper?

I ask this because I cant find any source yet commenting upon the relation of covariant/contravariant index of a component with respect to its designated column/row number.

Answer 1

Hi Jules,

Yes. If V is a finite-dimensional vector space of dimension n, then the space of (2,0), (1,1), and (0,2) tensors are all n^2-dimensional and hence all isomorphic to the vector space of n x n matrices (recall that all n^2-dimensional vector spaces are isomorphic to R^{n^2} and hence to each other). Concretely, let {e_1,...e_n} be an ordered basis for V and {ε^1,...,ε^n} the corresponding dual basis for V* defined by ε^i(e_j)=δ^i_j (where δ^i_j is the Kronecker delta symbol). Then a (0,2)-tensor is an element of V* ⊗ V*, a basis for which is given by the set {ε^i ⊗ ε^j}, where i and j both range from 1 to n. With respect to this basis, we write a tensor A as

A= Σ A_{ij} ε^i ⊗ ε^j (where we sum over both i and j from 1 to n). An explicit isomorphism from V* ⊗ V* to the vector space of n x n matrices is then given by taking A_{ij} to be the (i,j)-entry of a matrix.

I hope this helps!