Continuity of closest rank-p approximations to matrices

In signal processing, the singular value decomposition and rank characterization of matrices play prominent roles. The mapping which associates with any complex m × n matrix X its closest rank-p approximation X^(p)need not be continuous. When the pth and the (p + 1)st singular values of X are equal, this mapping maps, in fact, a matrix to a set of matrices. Furthermore, an example is given to show that large errors in computing X^(p)can be expected when σpis sufficiently close to σp+1. It is finally shown that this mapping is closed in the sense of Zangwill. The property of closedness is an essential assumption of a global convergence proof for algorithms involving this mapping (e.g., see [1]).