  Symmetrizable matrices are those which are symmetric when multiplied by a
diagonal matrix with positive entries. The Cauchy interlace theorem states that
the eigenvalues of a real symmetric matrix interlace with those of any
principal submatrix (obtained by deleting a row-column pair of the original
matrix). In this paper we extend the Cauchy interlace theorem for symmetric
matrices to this large class, called symmetrizable matrices. This extension is
interesting by the fact that in the symmetric case, the Cauchy interlace
theorem together with the Courant-Fischer minimax theorem and Sylvester's law
of inertia, each one can be proven from the others and thus they are
essentially equivalent. The first two theorems have important applications in
the singular value and eigenvalue decompositions, the third is useful in the
development and analysis of algorithms for the symmetric eigenvalue problem.
Consequently various and several applications whom are contingent on the
symmetric condition may occur for this large class of not necessary symmetric
matrices and open the door for many applications in future studies. We note
that our techniques are based on the celebrated Dodgson's identity.
