Consider a matrix $A\in\mathbb{R}^{n\times n}$. One of the eigenvalues of $A$ is zero and all the others are positive. Suppose $w\in\mathbb{R}^n$ is an eigenvector with the zero eigenvalue, i.e, $Aw=0$ Suppose $A$ is a function of time $t$. Hence the time derivative of $A$ and $w$ satisfy $(Aw)'=\dot{A}w+ A\dot{w}=0$ Has anybody encountered similar problems as below: if $\|\dot{A}\|$ is sufficiently small, under what kind of conditions $\|\dot{w}\|$ is also very small? Or in other words, if $\|\dot{A}\|$ is bounded from upper, when is $\|\dot{w}\|$ also bounded from upper?
PS: I omit some specifics of $A$ above. In case you may be familiar with graph theory, I am considering a transpose of a Laplacian matrix $A=L^T$. From graph theory, if the underlying graph is strongly connected, the Laplacian has a positive left eigenvector with the zero eigenvalue. By positive eigenvector, I mean all elements of the eigenvvector are positive.