Matrix representation of linear equation

To generate all force constants, we will create global matrices of force ( $\mathbf{F}$), displacements ( $\mathbf{U}$) and force constants ( $\mathbf{P}$). These are related each other as follows,

$$\mathbf{F} = - \mathbf{U} \cdot \mathbf{P}.$$ (3.2)

To explain the matrix representation of linear equation, first we will see solving technique of a force constant matrix between a displaced atom and forces on an atom in supercell.