For matrices that might be rank deficient it is common to incorporate pivoting in Householder QR factorization of A $\in$ $\Re^{mxn}$ (m $\geq$ n). Let $A^{(k)}$ denote the matrix at the start of the kth stage of the reduction (thus $A^{(1)}$ = A) and let $a_{j}^{(k)}$ denote the jth column of $A^{(k)}$. The column pivoting strategy excanges columns at the start of the kth stage to ensure that
||$a_{k}^{(k)}$(k : m)||$_{2}$ = $\max_{\substack{j \geq k}}$ ||$a_{j}^{(k)}$ (k : m)||$_{2}$

In other words, this strategy maximizes the 2-norm of the active part of the pivot column over all potential pivot columns.

a) Show that the R factor produced by QR factorization with column pivoting satisfies $r_{kk}^{2}$ $\geq$ $\sum_{i=k}^{j}$$r_{ij}^{2}$ , j = k + 1 : n, k = 1 : n.

b) What can be deduced about the ordering of the |$r_{ii}$| ?

c) If A has rank r < n, what is the form of R ?

Any help for the first question (a)?

Thanks in advance



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.