SciCode / 70.3 / Write a function to compute the expansion coefficients h_(k) of the three…

Background
The eight Gell-Mann matrices are:
$$
\begin{array}{ll}
\lambda_{1} & =\left(\begin{array}{lll}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0
\end{array}\right), \lambda_{2} =\left(\begin{array}{ccc}
0 & -i & 0 \\
i & 0 & 0 \\
0 & 0 & 0
\end{array}\right) \\
\lambda_{3} & =\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 0
\end{array}\right), \lambda_{4}=\left(\begin{array}{lll}
0 & 0 & 1 \\
0 & 0 & 0 \\
1 & 0 & 0
\end{array}\right) \\
\lambda_{5} & =\left(\begin{array}{ccc}
0 & 0 & -i \\
0 & 0 & 0 \\
i & 0 & 0
\end{array}\right), \lambda_{6}=\left(\begin{array}{lll}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{array}\right) \\
\lambda_{7} & =\left(\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & -i \\
0 & i & 0
\end{array}\right), \lambda_{8}=\frac{1}{\sqrt{3}}\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & -2
\end{array}\right)
\end{array}
$$

Each of the coefficient can be computed as the following:

$$
\begin{array}{c|l}
h_{1} & \operatorname{Re}\left[\left(\mathrm{H}\right)_{12}\right] \\
h_{2} & -\operatorname{Im}\left[\left(\mathrm{H}\right)_{12}\right] \\
h_{3} & \frac{1}{2}\left[\left(\mathrm{H}\right)_{11}-\left(\mathrm{H}\right)_{22}\right] \\
h_{4} & \operatorname{Re}\left[\left(\mathrm{H}\right)_{13}\right] \\
h_{5} & -\operatorname{Im}\left[\left(\mathrm{H}\right)_{13}\right] \\
h_{6} & \operatorname{Re}\left[\left(\mathrm{H}\right)_{23}\right] \\
h_{7} & -\operatorname{Im}\left[\left(\mathrm{H}\right)_{23}\right] \\
h_{8} & \frac{\sqrt{3}}{6}\left[\left(\mathrm{H}\right)_{11}+\left(\mathrm{H}\right)_{22}-2\left(\mathrm{H}\right)_{33}\right] \\
\end{array}
$$

As an example, the first coefficient h₁ is the real part of the number in the first row and the second column of the 3×3 Hamiltonian.

Background
The eight Gell-Mann matrices are:
$$
\begin{array}{ll}
\lambda_{1} & =\left(\begin{array}{lll}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0
\end{array}\right), \lambda_{2} =\left(\begin{array}{ccc}
0 & -i & 0 \\
i & 0 & 0 \\
0 & 0 & 0
\end{array}\right) \\
\lambda_{3} & =\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 0
\end{array}\right), \lambda_{4}=\left(\begin{array}{lll}
0 & 0 & 1 \\
0 & 0 & 0 \\
1 & 0 & 0
\end{array}\right) \\
\lambda_{5} & =\left(\begin{array}{ccc}
0 & 0 & -i \\
0 & 0 & 0 \\
i & 0 & 0
\end{array}\right), \lambda_{6}=\left(\begin{array}{lll}
0 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{array}\right) \\
\lambda_{7} & =\left(\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & -i \\
0 & i & 0
\end{array}\right), \lambda_{8}=\frac{1}{\sqrt{3}}\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & -2
\end{array}\right)
\end{array}
$$

Each of the coefficient can be computed as the following:

$$
\begin{array}{c|l}
h_{1} & \operatorname{Re}\left[\left(\mathrm{H}\right)_{12}\right] \\
h_{2} & -\operatorname{Im}\left[\left(\mathrm{H}\right)_{12}\right] \\
h_{3} & \frac{1}{2}\left[\left(\mathrm{H}\right)_{11}-\left(\mathrm{H}\right)_{22}\right] \\
h_{4} & \operatorname{Re}\left[\left(\mathrm{H}\right)_{13}\right] \\
h_{5} & -\operatorname{Im}\left[\left(\mathrm{H}\right)_{13}\right] \\
h_{6} & \operatorname{Re}\left[\left(\mathrm{H}\right)_{23}\right] \\
h_{7} & -\operatorname{Im}\left[\left(\mathrm{H}\right)_{23}\right] \\
h_{8} & \frac{\sqrt{3}}{6}\left[\left(\mathrm{H}\right)_{11}+\left(\mathrm{H}\right)_{22}-2\left(\mathrm{H}\right)_{33}\right] \\
\end{array}
$$

As an example, the first coefficient $h_1$ is the real part of the number in the first row and the second column of the $3 \times 3$ Hamiltonian.

Write a function to compute the expansion coefficients h_(k) of the three neutrino oscillation Hamiltonian as defined in the previous prompt () if we expand the Hamiltonian in SU(3) using the expression H=h₀1+h_(k)λ^(k), where 1 is the identity matrix, h_(k) is the expansion coefficient, and λ^(k) is the kth Gell-Mann matrix. Consider all the terms from k=1 to k=8.

Write a function to compute the expansion coefficients $h_k$ of the three neutrino oscillation Hamiltonian as defined in the previous prompt () if we expand the Hamiltonian in SU(3) using the expression $\mathbb{H}=h_{0} \mathbb{1}+h_{k} \lambda^{k}$, where $\mathbb{1}$ is the identity matrix, $h_k$ is the expansion coefficient, and $\lambda^{k}$ is the kth Gell-Mann matrix. Consider all the terms from $k=1$ to $k=8$.