Abstract
The URV and ULV decompositions are promising alternatives to the singular value decomposition for determining the numerical rank k of an m x n matrix and approximating its fundamental numerical subspaces whenever k ≈ min (m, n). This chapter proves general a posteriori bounds for assessing the quality of the subspaces obtained by two-sided orthogonal decompositions (such as the ULV and URV decompositions). The chapter shows that the quality of the subspaces obtained by the URV or ULV algorithm depends on the quality of the condition estimator and not on a gap condition.
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