2010/12/17 by Ricardo Reis da Silva, da Silva, Ricardo Reis
Computer Science · Mathematics · #Advanced Topics in Algebra #FOS: Mathematics #Mathematics and Applications #Matrix Theory and Algorithms #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1012.3835
openalex publication_date 2010/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a right eigenvector x and a left eigenvector y associated with the same eigenvalue of a matrix A, there is a Hermitian positive definite matrix H for which y=Hx. The matrix H defines an inner product and consequently also a field of values. The new generalized field of values is always the convex hull of the eigenvalues of A. Moreover, it is equal to the standard field of values when A is normal and is a particular case of the field of values associated with non-standard inner products proposed by Givens. As a consequence, in the same way as with Hermitian matrices, the eigenvalues of non-Hermitian matrices with real spectrum can be characterized in terms of extrema of a corresponding generalized Rayleigh Quotient.