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:<math> \mathbf{A}_{jk} = \mathbf{A}_{kj} \quad \text{for all } j, k </math>
Each dipole has three vector components (<math>x</math>, <math>y</math>, <math>z</math>), so we can rearrange the unknown vector <math>\mathbf{P}</math> by grouping all x-components together, then y-components, then z-components:
<math>
\mathbf{P} =
\begin{bmatrix}
\mathbf{P}_x \\
\mathbf{P}_y \\
\mathbf{P}_z
\end{bmatrix}, \quad \text{where} \quad
\mathbf{P}_x =
\begin{bmatrix}
P_{1x} \\
P_{2x} \\
\vdots \\
P_{Nx}
\end{bmatrix}, \quad
\mathbf{P}_y =
\begin{bmatrix}
P_{1y} \\
P_{2y} \\
\vdots \\
P_{Ny}
\end{bmatrix}, \quad
\mathbf{P}_z =
\begin{bmatrix}
P_{1z} \\
P_{2z} \\
\vdots \\
P_{Nz}
\end{bmatrix}
</math>
Similarly, the incident field can be grouped as:
<math>
\mathbf{E}^{\mathrm{inc}} =
\begin{bmatrix}
\mathbf{E}_x^{\mathrm{inc}} \\
\mathbf{E}_y^{\mathrm{inc}} \\
\mathbf{E}_z^{\mathrm{inc}}
\end{bmatrix}
</math>
Because the system is linear, we can equivalently rewrite it in block matrix form, that describe how the <math>\beta</math>-component of polarization affects the <math>\alpha</math>-component of the resulting field:
<math>
\begin{bmatrix}
\mathbf{A}_{xx} & \mathbf{A}_{xy} & \mathbf{A}_{xz} \\
\mathbf{A}_{yx} & \mathbf{A}_{yy} & \mathbf{A}_{yz} \\
\mathbf{A}_{zx} & \mathbf{A}_{zy} & \mathbf{A}_{zz}
\end{bmatrix}
\begin{bmatrix}
\mathbf{P}_x \\
\mathbf{P}_y \\
\mathbf{P}_z
\end{bmatrix}
=
\begin{bmatrix}
\mathbf{E}_x^{\mathrm{inc}} \\
\mathbf{E}_y^{\mathrm{inc}} \\
\mathbf{E}_z^{\mathrm{inc}}
\end{bmatrix}
</math>
Each matrix-vector multiplication <math>\mathbf{A}_{\alpha\beta} \mathbf{P}_\beta</math> can be computed as a convolution when the dipoles are arranged on a regular grid, allowing the use of Fast Fourier Transforms (FFTs) to accelerate the solution.
The expanded form of the equations is:
<math>
\begin{aligned}
\mathbf{A}_{xx} \mathbf{P}_x + \mathbf{A}_{xy} \mathbf{P}_y + \mathbf{A}_{xz} \mathbf{P}_z &= \mathbf{E}_x^{\mathrm{inc}} \\
\mathbf{A}_{yx} \mathbf{P}_x + \mathbf{A}_{yy} \mathbf{P}_y + \mathbf{A}_{yz} \mathbf{P}_z &= \mathbf{E}_y^{\mathrm{inc}} \\
\mathbf{A}_{zx} \mathbf{P}_x + \mathbf{A}_{zy} \mathbf{P}_y + \mathbf{A}_{zz} \mathbf{P}_z &= \mathbf{E}_z^{\mathrm{inc}}
\end{aligned}
</math>
==Fast Fourier Transform for fast convolution calculations==
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