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 n \times n matrices \{ A1, . . . ,Am\} , 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.