Say I find the eigenvalues of some matrix $A$ and then use kernel computation to find an eigenbasis for each eigenvalue. Then $B = S^{-1}AS$, where $S$ is composed of the eigenbasis of each of the eigenvalues. Does the order in which I arrange the eigenbasis in $S$ matter? For example, say I find some eigenvalues $\lambda_1$ and $\lambda_2$ with corresponding eigenvector $v$ and $w$, respectively. Then does it matter if I say that $S = vw$ or $wv$?
It changes my diagonal matrix, but its still diagonal no?