ORTHOGONALIZATION VIA DEFLATION : A MINIMUM NORM APPROACH FOR LOW-RANK APPROXIMATIONS OF A MATRIX

In this paper we introduce a new orthogonalization method. Given a real $m \times n$ matrix $A$, the new method constructs an SVD-type decomposition of the form $A = \hat U\hat\Sigma\hat V^T$. The columns of $\hat U$ and $\hat V$ are orthonormal, or nearly orthonormal, while $\hat\Sigma$ is a diagon...

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Bibliographic Details
Published in:SIAM journal on matrix analysis and applications 2008, Vol.30 (1), p.236-260
Main Author: DAX, Achiya
Format: Article
Language:English
Subjects:
Algebra
Approximation
Decomposition
Deflation
Eigenvalues
Eigenvectors
Equality
Exact sciences and technology
Linear and multilinear algebra, matrix theory
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Sciences and techniques of general use
Sequences, series, summability
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Summary:In this paper we introduce a new orthogonalization method. Given a real $m \times n$ matrix $A$, the new method constructs an SVD-type decomposition of the form $A = \hat U\hat\Sigma\hat V^T$. The columns of $\hat U$ and $\hat V$ are orthonormal, or nearly orthonormal, while $\hat\Sigma$ is a diagonal matrix whose diagonal entries approximate the singular values of $A$. The method has three versions: a "left-side" orthogonalization scheme in which the columns of $\hat U$ constitute an orthonormal basis of Range$(A)$, a "right-side" orthogonalization scheme in which the columns of $\hat V$ constitute an orthonormal basis of Range$(A^T)$, and a third version in which both $\hat U$ and $\hat V$ have orthonormal columns, but the decomposition is not exact. The new decompositions may substitute the SVD in many applications.
ISSN:0895-4798
1095-7162
DOI:10.1137/060656401