When is QR factorization naturally rank revealing?

Mark A.G. Smith; Ian K. Proudler

doi:10.1117/12.325683

2 October 1998 When is QR factorization naturally rank revealing?

Mark A.G. Smith, Ian K. Proudler

Proceedings Volume 3461, Advanced Signal Processing Algorithms, Architectures, and Implementations VIII; (1998) https://doi.org/10.1117/12.325683
Event: SPIE's International Symposium on Optical Science, Engineering, and Instrumentation, 1998, San Diego, CA, United States

Abstract

Taking the QR factorization of the covariance matrix M equals X^HX raised to increasing integer powers is shown to be equivalent to the process of Orthogonal Iteration and to converge upon a diagonal matrix with the eigenvalues of M raised to the corresponding power as its diagonal entries. This is a consequence of M being Hermitian. In addition, whereas the eigenvalues of a matrix are not in general rank-revealing, the eigenvalues of M ar as they are the squares of the singular values of X. In this way the Row-Zeroing approach to rank- revealing QR factorization is not longer defeated by the rank-deficient matrix due to Kahan. A connection is also noted with Chan's RRQR algorithm and a physical interpretation is developed for the function of Chan's permutation matrix. In practice, just a few steps of Orthogonal Iteration coupled with Row-Zeroing appears to be a very effective means of estimating the rank and signal subspace. The analytical error bounds upon the subspace estimate are much improved and as a consequence the diagonal value spectrum is sharpened, making thresholding easier and the R matrix much more naturally rank revealing. Hence it is sufficient to perform Row-Zeroing merely upon M or M². As a result, insight is also provided into the LMI method favored by Nickel.

Citation Download Citation

Mark A.G. Smith and Ian K. Proudler "When is QR factorization naturally rank revealing?", Proc. SPIE 3461, Advanced Signal Processing Algorithms, Architectures, and Implementations VIII, (2 October 1998); https://doi.org/10.1117/12.325683

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