Solving the problem of simultaneous diagonalization of complex symmetric matrices via congruence

We provide a solution to the problem of simultaneous $diagonalization$ $via$ $congruence$ of a given set of $m$ complex symmetric $ntimes n$ matrices ${A_{1},ldots,A_{m}}$, by showing that it can be reduced to a possibly lower-dimensional problem where the question is rephrased in terms of the classical problem of simultaneous $diagonalization$ $via$ $similarity$ of a new related set of matrices. We provide a procedure to determine in a finite number of steps whether or not a set of matrices is simultaneously diagonalizable by congruence. This solves a long standing problem in the complex case.