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where I~I, ~r, and W are matrices with orthonormal columns, while L and L1 are lower
triangular matrices. In particular,
L= G E '
where LK is K x K with ~min(LK)~ aK, and IIGII~- IIEII~ ~ a~+ 1 +...+~; ~min(LK)
denotes the smallest singular value of LK and IIGIIF (and I]EIIF)is the Frobenious norm
of G (and E). By analogy with (20), we can then write
i.e., the rank K of HR -1 is displayed in the matrix L in that LK is well conditioned, and
that [[(~'I]F and I]EI]F are small. In addition, 01 = [51,..., IlK] (and 02 = [UK+I,'", UM])
represents approximately the same space ...