Spectral decomposition of real circulant matrices

AbstractThis paper presents spectral decompositions, i.e., eigendecompositions and singular value decompositions of four types of real circulant matrices. Right and left circulants (whose elements topple from right to left or from left to right, respectively) as well as skew right and skew left circulants (whose elements change their sign when toppling) are analyzed.The inherent periodicity of circulant matrices means that they are closely related to Fourier analysis and group theory. This relationship is utilized in the spectral decompositions of this paper

analysis of continuous methods for unconstrained optimization and their discretizations

abstract: the well-known difficulties with the treatment of ill-conditioned unconstrained optimization problems can be explained by analogous difficulties with stiff differential equations. this observation provides the basis for an analysis of optimization problems and reveals new classes of optimization methods. this paper is primarily theoretic, a subsequent paper will be devoted to practical aspects of the proposed methods.