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 $\dot\lambda(0)=x^*A(0)x$ \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.
