On Joint Eigenvalues of Commuting Matrices

A spectral radius formula for commuting tuples of operators has been proved in recent years. We obtain an analog for all the joint eigenvalues of a commuting tuple of matrices. For a single matrix this reduces to an old result of Yamamoto. 1 Introduction, formulation of the result Let T = (T 1 ; : : : ; T s ) be an s-tuple of complex d \Theta d-matrices. The joint spectrum oe pt (T ) is the set of all points = ( 1 ; : : : ; s ) 2 C s (called joint eigenvalues) for which there exists a nonzero vector x 2 C d (called joint eigenvector) satisfying T j x = j x for j = 1; : : : ; s: (1) If the T 0 i s are commuting then oe pt (T ) 6= ;. The joint spectrum can be read off the diagonal of the common triangular form: There exists a unitary d \Theta d - matrix U such that U H T j U = 0 B B B B @ (j) 1 : : : : : : : : : 0 (j) 2 : : : : : : 0 0 . . . . . . 0 0 0 (j) d 1 C C C C A for j = 1; : : : ; s: (2) Indian Statistical Institute, Delhi centre, 7, SJS Sansanwal Ma..