  An equivalence between attainability of simultaneous diagonalization (SD) and
hidden convexity in quadratically constrained quadratic programming (QCQP)
stimulates us to investigate necessary and sufficient SD conditions, which is
one 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 SD
condition for any two real symmetric matrices and offer a necessary and
sufficient SD condition for any finite collection of real symmetric matrices
under the existence assumption of a semi-definite matrix pencil. Moreover, we
apply our SD conditions to QCQP, especially with one or two quadratic
constraints, to verify the exactness of its second-order cone programming
relaxation and to facilitate the solution process of QCQP.
