In the physical sciences, the scientist mostly tries to put the problem in a form that is easily solvable. One such form is the eigenvector expansion where one solves a simpler problem and assumes that the more difficult problem can be decomposed in a sum of the simpler solutions.

This kind of problem generally involves solving the eigenvalue problem of a particular matrix. Assuming that you know how to do that, this is all fine and well. However, it is also common to use orthogonality relations to simplify the solution. For a general matrix $A$, the resulting eigenvectors need not be orthogonal. That's a problem. Let's look for a solution.

Adopting the braket notation and denoting an eigenvector by $|e_i\rangle$, we seek new vectors such that
$$ \langle f^i|e_j\rangle = \delta^i_j.$$
To see how we can compute those vectors, consider the original
eigenvalue problem
$$ \label{eq:eigenvalue}A|e_i\rangle = \lambda_i|e_i\rangle .$$
We will use the fact that both sets of eigenvectors are complete
so that there exists a closure relation
$$ \sum_i |e_i\rangle\langle f^i| = 1.$$
Now, pre-multipling \eqref{eq:eigenvalue} by $\langle f^j|$ and post-multiplying by
$\langle f^i|$ and summing over the index $i$, we get
$$ \sum_i \langle f^j|A|e_i\rangle\langle f^i| = \sum_i \lambda_i \langle f^j|e_i\rangle\langle f^i|.$$
Using our orthogonality relations on the right-hand side of this last equation and the closure
relation on the left-hand side, we get
$$ \langle f^j| A = \sum_i \lambda_i \langle f^i| \delta^j_i. $$
The Dirac delta makes the sum disappear as usual and we have
$$ \langle f^j| A = \lambda_j \langle f^j|. $$
In other words, the $\langle f^j|$ are the row eigenvectors of
$A$ and have the same eigenvalues as the set $|e_i\rangle$!

So we find that these new "contravariant" vectors are actually the *left eigenvectors* of the matrix $A$. Moreover, the decomposition of any vector in the original set of covariant vectors is given by the dot product with the vectors in the dual basis. Say we have a vector $|v\rangle$, its decomposition is given by
$$ |v\rangle = \sum_i \langle f^i|v\rangle|e_i\rangle .$$

This eigendecomposition is incredibly useful in solving differential equations, smoothing numerically unstable solutions...