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An equivalence between attainability of simultaneous diagonalization (SD) andhidden convexity in quadratically constrained quadratic programming (QCQP)stimulates us to investigate necessary and sufficient SD conditions, which isone of the open problems posted by Hiriart-Urruty (SIAM Rev., 49 (2007), pp.255-273) nine years ago. In this paper we give a necessary and sufficient SDcondition for any two real symmetric matrices and offer a necessary andsufficient SD condition for any finite collection of real symmetric matricesunder the existence assumption of a semi-definite matrix pencil. Moreover, weapply our SD conditions to QCQP, especially with one or two quadraticconstraints, to verify the exactness of its second-order cone programmingrelaxation and to facilitate the solution process of QCQP.

Simultaneous Diagonalization of Matrices and its Applications in Quadratically Constrained Quadratic Programming