# Effects of elementary row operation on condition number

How does any elementary row operation on a matrix affect the condition number?

Can an ill conditioned matrix be improved by just some elementary row operations?

Can I improve the accuracy of solving linear system by some row operations?

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'Pre-conditioning' by multiplying by a diagonal matrix is a fairly common way of attempting to improve the condition number of a matrix. – copper.hat May 25 '12 at 5:57

If we write down the svd of the matrix $A$ as $A = U \Sigma V^T$ and if $P$ is a permutation matrix, then $P \times A = (P U) \Sigma V^T$.
$PU$ is again a unitary matrix. Hence, the singular values of $A$ and $PA$ are the same.
However, if your row operations include scaling and adding scalar multiples of rows then it definitely does affects the condition number. In fact, thats what a pre-conditioner for a matrix does. When you pre-multiply a matrix $A$ by another matrix $T$, you are essentially scaling and adding scalar multiple of rows with each other. This clearly affects the condition number. A degenerate example would be if you pre-multiply $A$ by $A^{-1}$, then the condition number of the resulting matrix is $1$.
Scaling does affect the condition number. The condition number of $\mathbb{diag(1,1)}$ is $1$, but when multiplied by $\mathbb{diag(1,2)}$, the condition number becomes $2$. – copper.hat May 25 '12 at 5:52