Generalized Rayleigh-Quotient Formulas for the Real Parts, Imaginary Parts, and Moduli of the Eigenvalues of General Matrices

Abstract

In the present paper, generalized Rayleigh-quotient formulas for the real parts, imaginary parts, and moduli of the eigenvalues of general (not necessarily diagonalizable) matrices are derived by using quotients of the form $(Au,v)/(u,v)$ instead of $(Au,u)/(u,u)$. These formulas are new and correspond to similar formulas for diagonalizable matrices obtained recently. Numerical examples underpin the theoretical findings. We point out that, in the case of general matrices, the principal vectors of largest stage of matrix $A^{\ast}$ take over the role of the eigenvectors in the case of diagonalizable matrices. So, even though the formulas in both cases look very similar, the result is somehow unexpected and surprising.

Keywords

• general (not necessarily diagonalizable) matrix
• real and imaginary parts of eigenvalues; generalized rayleigh-quotient formulas
• generalized numerical range
• system matrix of a linear dynamical problem