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Decomposition of an updated correlation matrix via hyperbolic transformation. (English) Zbl 1090.65046
Given a rectangular, in general complex, \(m\times n\) matrix \(X\), the problem is to compute singular values of \(X\) and of its update \(X^{\text{new}}\) obtained by omission of certain columns—\(m\times p\) matrix \(Y\)—and addition of new ones— \(m\times k\)-matrix \(Z\). The classical way is to use the singular decomposition (SVD) algorithm based on the lower triangularization of \(X,\;Y\) and \(Z\).
In the paper, another method is presented for computing singular values of \(X^{\text{new}}\), namely the hyperbolic SVD based on the hyperbolic Givens and Householder transformations matrices. Their properties have been proved in [D. Janovská and G. Opfer, Numer. Linear Algebra 8, 127–146 (2001; Zbl 1051.65045)] and are shortly reviewed here. Then the existence conditions of the hyperbolic SVD are recalled and an algorithm proposed using sequentially the Householder and Givens transformations. Finally, some applications of the algorithm are mentioned.
Reviewer: Ivan Saxl (Praha)
65F25 Orthogonalization in numerical linear algebra
65F30 Other matrix algorithms (MSC2010)
Zbl 1051.65045
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