DETERMINANT SPECTRUM: A GENERALIZATION OF EIGENVALUES

Abstract

In this paper we introduce a generalization of eigenvalues called determinant spectrum for an element in the matrix algebra. The importance of determinant spectrum is reflected from the application of lemniscates. Determinant spectrum is also useful in various other fields of mathematics, especially in the numerical solution of matrix equations. Determinant spectrum shares some properties of eigenvalues, at the same time, it has many properties that are different from the properties of eigenvalues. In this paper we study about the linear map preserving determinant spectrum on the matrix algebra. We prove that the linear map preserving determinant spectrum on the matrix algebra preserves eigenvalues and their multiplicity. We also prove an analogue of the Spectral Mapping Theorem for determinant spectrum in the matrix algebra. The usual Spectral Mapping Theorem is proved as a special case of this result. The results developed are illustrated with examples and pictures using matlab.