QR and permutation matrix

[Han]

The QR() command returns the QR factorization of a matrix and includes a permutation matrix. I cannot seem to find an example of a matrix for which the QR factorization returns a permutation matrix that is non-identity. Is anyone able to find such a case? From the looks of it, the QR() command does not appear to do any pivoting (the diagonals of R are not in non-increasing order). For example:

M2:=[[1,2],[3,5],[-1,7],[2,-1]]
QR(M2);

returns

Code:

[
  [0.258198889747,0.169657961696,0.94436252598,−0.112822554889],
  [0.774596669241,0.393298002112,−0.326894720531,−0.372097464682],
  [−0.258198889747,0.871424985073,−3.63216356146e−2,0.415490755024],
  [0.516397779494,−0.23906349148,0,0.82230285198]
],
[
  [3.87298334621,2.06559111798],
  [0,8.64484432094],
  [0,0],
  [0,0]
],
[
  [1,0,0,0],
  [0,1,0,0],
  [0,0,1,0],
  [0,0,0,1]
]

So is there no pivoting? And if not, then it appears the P matrix is superfluous.

[parisse]

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

[Han]

I actually do not need the permutation matrix. I was asking because I wanted to use the QR factorization for solving a system and if the permutation matrix is non-identity then it would have required a few extra lines of code.

Is this the same case for LU factorizations (I.e safe to ignore the 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.

[parisse]

I see, it is probably like total pivoting vs partial pivoting. Partial pivoting is used in LU and QR, but in the LU case this adds a generically non trivial permutation matrix (PA=LU, P^-1*L would not be triangular) while in the QR case the permutation is absorbed by Q.