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We consider orthogonal transformations of arbitrary square matrices to a formwhere all diagonal entries are equal. In our main results we treat thesimultaneous transformation of two matrices and the symplectic orthogonaltransformation of one matrix. A relation to the joint real numerical range isworked out, efficient numerical algorithms are developped and applications tostabilization by rotation and by noise are presented.

Simultaneous hollowisation, joint numerical range, and stabilization by noise