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

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