QR and permutation matrix

[Han]

(10-26-2015 05:04 PM)parisse Wrote:  There is indeed no need to have a permutation matrix, you can check that in giac in vecteur.cc in qr_ortho, the idn matrix is here for compatibility. There is pivoting in the sense that to reduce a given column the line with highest absolute value is choosen.

I am not sure what you mean by this. If
\[ A = \begin{bmatrix}
a_{1} & a_{2} & \dotsm & a_{n} \\
\end{bmatrix} \]
then the first step is to swap columns (if necessary) so that the column column \( a_1 \) is replaced with the column having largest norm? And then a Householder reflection is applied? (And similarly for submatrices) Is that what you meant by pivoting?

Quote:Can you explain what is the reason behind having a permutation matrix?

QR factorization with pivoting gives R with diagonal terms in non-decreasing order. This is useful for factoring rank-deficient matrices.