Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.