Relationship Between the Stiffly Weighted Pseudoinverse and Multi-Level Constrained Rseudoinverse

Abstract

It is known that for a given matrix $A$ of rank $r$, and a set $D$ of positive diagonal matrices, $\sup_{W\in D}||(W^{\frac{1}{2}}A)^†W^{\frac{1}{2}}||_2=(\min_i \sigma_+(A^{(i)})^{-1}$, in which $(A^{(i)})$is a submatrix of A formed with $r = (\rm{rank}(A))$ rows of $A$, such that $(A^{(i)})$ has full row rank $r$. In many practical applications this value is too large to be used. 

In this paper we consider the case that both $A$ and $W(\in D)$ are fixed with $W$ severely stiff. We show that in this case the weighted pseudoinverse $W^{\frac{1}{2}}A)^†W^{\frac{1}{2}}$ is close to a multi-level constrained weighted pseudoinverse therefore $||(W^{\frac{1}{2}}A)^†W^{\frac{1}{2}}||_2$ is uniformly bounded. We also prove that in this case the solution set the stiffly weighted least squares problem is close to that of corresponding multi-level constrained least squares problem.