In some unrigorous physics texts the following "inverse" is used informally:
$$\sum_{k,l=1}^n A_{ijkl} x_{kl} = b_{ij} \qquad \Rightarrow x_{kl} = \sum_{i,j=1}^n (A_{ijkl})^{-1} b_{ij}, \qquad (*)$$
where $A_{ijkl}$ are the componentes of a fourth order tensor, and $x_{kl}$ and $b_{ij}$ the components of two second order tensors. Although often there does not seem to be a general definition of this inverse (nor are the conditions under which such an object exists explicit). Reviewing the definition of adjugate matrix $\mathbf{A} = \text{adj}(\mathbf{a}) = (A_{ij})$, we have that:
$$ \sum_j a_{ij} x_j = b_i \Rightarrow \sum_k \sum_j A_{ki} a_{ij} x_j = \sum_k A_{ki} b_i \Rightarrow \det(\mathbf{a}) x_j = \sum_k A_{ki} b_i \Rightarrow \boxed{\mathbf{x} = \frac{\text{adj}(\mathbf{a})\mathbf{b}}{\det(\mathbf{a})}}$$
In this case the condition of existence of inverses is the condition that $\det(\mathbf{a}) \neq 0$ . But what can be said in the multilinear case, specifically, my questions are:
- For the case of fourth-order tensors, what would be the relevant generalization of the determinant that guarantees the existence of the inverse as in equation $(*)$?
- Given the above condition, how could the corresponding adjugate tensor be calculated?
- In the case of other orders, three, five, etc., what can be said in general about the existence of adjugate tensors and the conditions for the existence of "inverses" in the sense of equation X?