  We propose and develop a theory of Ferrers diagrams and their $q$-rook
polynomials solely based on their diagonals. We show that the cardinalities of
the diagonals of a Ferrers diagram are equivalent information to their rook
numbers, $q$-rook polynomials, and the rank distribution of matrices supported
on the diagram. Our approach is based on the concept of \textit{canonical form}
of a Ferrers diagrams, and on two simple diagram operations as the main proof
tools. In the second part of the paper we develop the same theory for symmetric
Ferrers diagrams, considering symmetric and alternating matrices supported on
them. As an application of our results, we establish some combinatorial
identities linking symmetric and alternating matrices, which do not appear to
have an obvious bijective proof, and which generalize some curious results in
enumerative combinatorics.
