Simultaneous Hollowization, Joint Numerical Range, and Stabilization by Noise

We consider orthogonal transformations of arbitrary square matrices to a form where all diagonal entries are equal. In our main results we treat the simultaneous transformation of two matrices and the symplectic orthogonal transformation of one matrix. A relation to the joint real numerical range is worked out, efficient numerical algorithms are developed, and applications to stabilization by rotation and by noise are presented.