# do i have to normalize the matrix?

let say i have a matrix $$A= \begin{bmatrix} 1.00 & -28.00 & -268.00 & 13596.00 & -50165.00 \\ 0.00 & -96.00 & 2496.00 & -6048.00 & -70080.00 \\ -97.00 & 2524.00 & -5876.00 & -151260.00 & 44117.00 \\ 2.00 & -18584.00 & 628648.00 & -5858664.00 & 5101142.00 \\ -18435.00 & 626388.00 & -5887452.00 & 11104812.00 & -6009633.00 \\ \end{bmatrix}$$

Here I am trying to solve for linear systems of equations. I've used partial pivoting Gauss Elimination and QR householder decomposition to solve my problem. but my concern is that i got a better solution using PPGE compared to QR householder (knowing that QR householder is much stable said in literature).

So my concern is, is there a way so that I can improve my solutions? Is there any method that can give a better result? Just before I come across that normalizing the matrix as doing that way will help in reducing the condition number of the matrix (the condition of the matrix A is big, but eventually it still give a good result using the PPGE). So I googled and searched for normalizing the matrix, but mostly its for eigenvalues problem. So I'm a little bit confused here. Can I apply the same technique to my problem, means, to normalize a matrix, do I just divide every vector in a column with their say 1-norm?

Is it because the difference between the first element and the last element of A effect the answer? ouh, i've also tried QR with column pivoting, but the answer is not as good as QR Householder and PPGE.

I really appreciate any reply. TQ.

• Can you post results/ What does it mean better? – Moo Nov 29 '16 at 1:11
• @moo here is the result – olive Nov 29 '16 at 2:27
• sorry @moo. i mean when i'm doing the computation on some examples by PPGE, i got the exact result (or very near to the solution). but then when i use QR to solve the same example by PPGE before, the solution give a relative error. i mean shouldn't QR give a better result than PPGE? – olive Nov 29 '16 at 2:37