Eigenvalues of Composite Matrices

Because of the symmetry of the problems encountered in the applications of matrices to physics, electrical engineering and numerical analysis, it frequently turns out that the matrix to be considered is a composite matrix, that is, a matrix whose elements are matrices. For example, the matrix may be where A 1 and A 2 are square matrices of order n which need not commute. It is easy to prove that the eigenvalues of this matrix of order 2 n are the eigenvalues of the two matrices, A 1 + A 2 and A 1 − A 2 of order n .