A matrix eigenvalue–eigenvector equality for semi-simple eigenvalues

Abstract

For a complex square matrix, we present an eigenvalue–eigenvector equality for its semi-simple eigenvalue with a basis of the corresponding eigenspace under the condition that the eigenspace is orthogonal to eigenspaces or generalized eigenspaces corresponding to all other eigenvalues of the matrix. As a special case, we obtain a generalized eigenvector–eigenvalue-identity for the eigenvalue with an orthonormal basis of the eigenspace, which generalizes the well-known eigenvector–eigenvalue identity for a simple eigenvalue of normal matrices. We also give an application of the new formula to Jacobi matrices.