It's not specified at all because there is no such matrix - the e_i are vectors, not scalars. At best it's a convenient mnemonic abbreviation for something deci

It's not specified at all because there is no such matrix - the e_i are vectors, not scalars. At best it's a convenient mnemonic abbreviation for something decided on some other basis; that's like saying that the gradient is 'specified' by the 'vector' ∇ = <∂/∂x,∂/∂y,∂/∂z>.