Simultaneous diagonalization via congruence of $m$ real symmetric matrices and its implications in optimization

Let $\{C_1, C_2, \ldots, C_m\},~m\ge2$ be a collection of $n\times n$ real symmetric matrices. The objective of the paper is to offer an algorithm that finds a common congruence matrix $R$ such that $R^TC_iR$ is real diagonal for every $C_i;$ or reports none of such kind. The problem, referred to as the simultaneously diagonalization via congruence (SDC in short), seems to be of pure linear algebra at first glance. However, for quadratically constrained quadratic programming (QCQP), if the quadratic forms are SDC, their joint range set is a closed convex polyhedral cone, which opens the possibility to extend the classical $\mathcal{S}$-lemma for more than two symmetric matrices. In addition, under the SDC assumption of quadratic forms, QCQP can be recast in separable forms which is usually easier to tackle. It is thus important to have a standard procedure for determining whether or not the SDC property holds for the underlined quadratic optimization problem. Our result solves a long standing problem posed by Hiriart-Urruty in 2007.