I.3.7.2: Low Rank Approximation of a Matrix

Given an matrix we want to minimize over the matrices and the matrices . In other words, we find the projection of , in the Frobenius norm, on the set of matrices of rank less than or equal to .

The block relaxation iterations are or

It follows that and, thus, for all Thus increases to a limit less than or equal to the upper bound . Also and consequently converges to zero.

Now suppose , with nonsingular. Then , and thus In addition , and thus

Alternative