engleski

Two-sided hyperbolic singular value decomposition

The aim of this work is construction of a singular value decomposition of a given matrix A with regards to a hyperbolic scalar product induced by J (called two-sided SVD or, shortly, just 2HSVD), maintaining the properties similar to those of a traditional singular value decomposition. In the first chapter an overview of the thesis and known results in the indefinite linear algebra are given. Some of the plethora of the variations of the singular value decomposition are given. The third chapter brings the main theoretical results: 2HSVD itself, its variant for J-nonnegative matrices, relation to a semidefinite J-polar decomposition and its properties similar or equal to those in Euclidean case. Two algorithms -- two- and one-sided Jacobi-like -- along with their numerical properties are discussed in the fourth chapter. The next chapter shows one application of 2HSVD to a generalization of a Zakrajšek-Vidav decomposition of block-unitary matrices. The final chapter discusses further generalizations of such decomposition to more general scalar product spaces.

SVD; hyperbolic scalar products

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