## Start from easy and low dimensional matrix:

$m\vec v=\vec f \longrightarrow \vec v=m^{-1}\vec f$

$\vec v= \left\{ \begin{matrix} \overline m_{11}&\overline m_{12}\\ \overline m_{21}&\overline m_{22} \end{matrix} \right\} \cdot \left\{ \begin{matrix} f_1\\ f_2 \end{matrix} \right\} =(\overline m_{11}f_1+\overline m_{12}f_2)\hat e_1+(\overline m_{21}f_1+\overline m_{22}f_2)\hat e_2$

Thus,

$v_i=\Sigma_j\overline m_{ij}f_j$

## For function:

$\frac{d^2y}{dx^2}+k^2y=f(x)$

operator:

$\hat O=\frac{d^2}{dx^2}+k^2$

$\hat Oy(x)=f(x)\longrightarrow y(x)=\hat O^{-1}f(x) ?$

$\frac{d}{dx}=\int \delta(x-x')\frac{d}{dx'}q\cdot dx'$

$\downarrow$

2 indexes

$y(x)=\int G(x,x')f(x')dx'$

$\downarrow$

$\hat OG(x,x')=\delta(x-x')$

## Conclusion:

Just the result of an infinite dimensional inverse problem.