A necessary and sufficient condition for simultaneous diagonalization of two hermitian matrices and its application

Abstract

We denote by F the field R of real numbers, the field C of complex numbers, or the skew field H of real quaternions, and by Fⁿ an n dimensional left vector space over F. If A is a matrix with elements in F, we denote by A^* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be hermitian if A = A^*, and we say that two n × n hermitian matrices A and B with elements in F can be diagonalized simultaneously if there exists a non singular matrix U with elements in F such that UAU^* and UBU^* are diagonal matrices. We shall regard a vector u ∈ F_n as a l × n matrix and identify a 1 × 1 matrix with its single element, and we shall denote by diag {A₁, …, A_m} a diagonal block matrix with the square matrices A₁, …, A_m lying on its diagonal.