I am currently learning about the Cauchy stress tensor which is a rank (0,2) contravariant tensor. Unless I am mistaken, this means that the tensor maps 2 dual vectors to a scalar. Now, in class we recently learned about the concept of principle stresses. These were described as the eigenvalues of the tensor with the associated eigenvectors. To find them, we diagonalized the matrix of the tensor and found the eigenvectors in the usual way from linear algebra. However, I don't understand what that even means. From linear algebra, I know that an eigenvector is a vector $v$ associated with a linear operator $L$ such that:
$Lv = \lambda v$
where $\lambda$ is called the eigenvalue.
The Cauchy stress tensor doesn't even map vectors but rather dual vectors. How can we pretend that the tensor is just a matrix (which itself is just a linear operator in a specific basis) and find eigenvalues and eigenvectors in the usual way?