I have a linear* problem which arises from an integro-differential system, and writes: $$ (\mathbf{I}+\lambda \mathbf{A})x = b $$ where $\mathbf{A}$ is a real full matrix, size $n\times n$, but is not symmetric and has some zero eigenvalues. I know what these eigenvalues correspond to.
When $\lambda$ increases (but is still much smaller than 1) the condition number of $\mathbf{I}+\lambda \mathbf{A}$ (let's call it $\mathbf{K}$) deteriorates extremely rapidly (to $10^{10}$ and more). Even with a direct solver (luckily $n=100$ is enough for me) the results become meaningless.
I have no certainty, but I believe this is linked with the zero eigenvalues of $\mathbf{A}$. Is there a way I could exploit my knowledge of the corresponding eigenvectors to regularize the numerical resolution?
(* note that this is actually the fixed-point linear problem that I am solving, the original problem has $\lambda=\lambda(x)$)
EDIT: $\mathbf{A}$ itself has indeed a very large condition number (numerically evaluated by numpy with L2-norm), $10^{19}$, with $\sigma_{\min} = 1.8e-07$ and $\sigma_{\max} = 3.2e11$ ($\sigma_{\min}$ should analytically be 0 of course, since as said above, $\mathbf{A}$ has a 0 eigenvalue). Its largest eigenvalue has modulus about 600, here is the complete spectrum for $n=102$