Is there any reference on whether expressions of mixed order tensors, for instance:
$$a = v_iT^{i}_{.j}u^j$$
which evaluate to the same scalar (irrespective of the order of these terms above) correspond exactly to:
$$a = \begin{bmatrix} v_1 & \cdots &v_n \end{bmatrix} \begin{bmatrix} T^1_1 &\cdots &T^1_n \\ \vdots & \ddots & \vdots \\T^n_1 & \cdots & T_n^n \end{bmatrix} \begin{bmatrix} u_1 \\ \vdots \\u_n \end{bmatrix}$$
if the contracted indices obey a particular order (covariant first, contravariant second, as the matrix product above or the other way around)?
In other expressions, such as those with cartesian tensors like: $$E = \frac{1}{2}\int I_{ij}\omega_i \omega_jdV$$ can we show that any matrix combination of these terms that give the resulting matrix in the correct dimensions (1x1 in this case) will result in the correct contraction/operation (given that we can raise and lower indices as we like, as I suppose the metric is simply the identity matrix)?
