### 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 spec****trum 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 Map****ping 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.**

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