A set of complex symmetric matrices is simultaneously diagonalizable by congruence exactly when the common kernel has the expected dimension and a related set of reduced matrices is simultaneously diagonalizable by similarity.
B., Potpourri of conjectures and open questions in nonlinear analysis and optimisation, SIAM Rev., 49, (2007), 255–273
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Solving the problem of simultaneous diagonalization of complex symmetric matrices via congruence
A set of complex symmetric matrices is simultaneously diagonalizable by congruence exactly when the common kernel has the expected dimension and a related set of reduced matrices is simultaneously diagonalizable by similarity.