The hyperbolic Schur decomposition

Abstract

We propose a hyperbolic counterpart of the Schur decomposition, with the emphasis on the preservation of structures related to some given hyperbolic scalar product. We give results regarding the existence of such a decomposition and research the properties of its block triangular factor for various structured matrices.Keywords: indefinite scalar products, hyperbolic scalar products, Schur decomposition, Jordan decomposition, quasitriangularization, quasidiagonalization, structured matrices 2000 MSC: 15A63, 46C20, 65F25 IntroductionThe Schur decomposition A = U T U * , sometimes also called Schur's unitary triangularization, is a unitary similarity between any given square matrix A ∈ C n×n and some upper triangular matrix T ∈ C n×n . Such a decomposition has a structured form for various structured matrices, i.e., T is diagonal if and only if A is normal, real diagonal if and only if A is Hermitian, positive (nonnegative) real diagonal if and only if A is positive (semi)definite and so on.Furthermore, the Schur decomposition can be computed in a numerically stable way, making it a good choice for calculating the eigenvalues of A (which are the diagonal elements of T ) as well as the various matrix functions (for more details, see [11]). Its structure preserving property allows to save time and memory when working with structured matrices. For example, computing the value of some function of a Hermitian matrix is reduced to working with a diagonal matrix, which involves only evaluation of the diagonal elements.Unitary matrices are very useful when working with the traditional Euclidean scalar product x, y = y * x, as their columns form an orthonormal basis of C n . However, many applications require a nonstandard scalar product which is usually defined by [x, y] J = y * Jx, where J is some nonsingular matrix, and many of these applications consider Hermitian or skew-Hermitian J. The hyperbolic Email address: vsego@math.hr (VedranŠego) scalar product defined by a signature matrix J = diag(j 1 , . . . , j n ) (j k ∈ {−1, 1}) arises frequently in applications. It is used, for example, in the theory of relativity and in the research of the polarized light. More on the applications of such products can be found in [10,13,14,17].The Euclidean matrix decompositions have some nice structure preserving properties even in nonstandard scalar products, as shown by Mackey, Mackey and Tisseur [16], but it is often worth looking into versions of such decompositions that respect the structures related with the given scalar product. There is plenty of research on the subject, i.e., hyperbolic SVD [17,24], J 1 J 2 -SVD [9], two-sided hyperbolic SVD [20], hyperbolic CS decomposition [8,10] and indefinite QR factorization [19].There are many advantages of using decompositions related to some specific, nonstandard scalar product, as such decompositions preserve structures related to a given scalar product. They can simplify calculation and provide a better insight into the structures of such structured matrices.In this paper w...

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