When is QR factorization naturally rank revealing?
Mark A. Smith, Ian K. Proudler · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1998
Taking the QR factorization of the covariance matrix M equals XHX 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.