The hyperbolic singular value decomposition and applications

Abstract

A new generalization of the singular value decomposition (SVD), the hyperbolic SVD, is advanced, and its existence is established under mild restrictions. The hyperbolic SVD accurately and efficiently finds the eigenstructure of any matrix that is expressed as the difference of two matrix outer products. Signal processing applications where this task arises include the covariance differencing algorithm for bearing estimation in sensor arrays, sliding rectangular windowing, and array calibration. Two algorithms for effecting this decomposition are detailed. One is sequential and follows a similar pattern to the sequential bidiagonal based SVD algorithm. The other is for parallel implementation and mimics Hestenes' SVD algorithm (1958). Numerical examples demonstrate that like its conventional counterpart, the hyperbolic SVD exhibits superior numerical behavior relative to explicit formation and solution of the normal equations. Furthermore, the hyperbolic SVD applies in problems where the conventional SVD cannot be employed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>