Theorem Let $t\mapsto A(t)$ a $C^1$-family of $n\times n$ matrices. If at $t=0$, $A(0)$ has a simple eigenvalue $\lambda_0$, then for $t$ small enough, $A(t)$ admits a simple eigenvalue $\lambda(t)$ that is a $C^1$-function of $t$ and is such that $\lambda(0)=\lambda_0$. In addition, its derivative at $t=0$ is given by $\dot\lambda(0)=x^*A(0)x$ \dot\lambda(0)=x^*\dot A(0)x$where$x$is the normalized ($\|x\|=1$) eigenvector associated with$\lambda_0$. Of course, there are infinite-dimensional versions of this statement. Both finite and infinite-dimensinal versions are used daily by hundreds of mathematicians. 1 A fundamental application is Theorem Let$t\mapsto A(t)$a$C^1$-family of$n\times n$matrices. If at$t=0$,$A(0)$has a simple eigenvalue$\lambda_0$, then for$t$small enough,$A(t)$admits a simple eigenvalue$\lambda(t)$that is a$C^1$-function of$t$and is such that$\lambda(0)=\lambda_0$. In addition, its derivative at$t=0$is given by$\dot\lambda(0)=x^*A(0)x$where$x$is the normalized ($\|x\|=1$) eigenvector associated with$\lambda_0\$.