eigen values and eigen vectors in case of matrixes and differential equations

Question

When I have an equation of the form $$a\frac{d^2x}{dt^2}+b\frac{dx}{dt}+c=0$$, then I can use Laplace transformation and transform it into $$as^2+bs+c=0$$ provided obviously$\quad x"(0)=0\quad \text{and}\quad x'(0)=0$. But the equation $as^2+bs+c=0$ is called the characteristic equation and the roots of the equation are called the eigen values $\lambda_1,\lambda_2$. Then we write the solution as $A\exp{\lambda_1t}+B\exp{\lambda_2t}$. I also know that if I have a vector space $\mathbb{V}\cap\mathbb{C}^2$, I can write $$H\left|x\right>=\lambda\left|x\right>$$ where the two $\lambda$ values are the eigen values of the eigen vectors $\left|x\right>$. Is there a correlation between the two, I mean can the polynomial $as^2+bs+c=0$ be represented in the form of a matrix/eigen vectors? Why do we call the roots of the characteristic equation the eigen values? What is the correlation? I'm sure I'm missing out on something.