<s>[INST] Matrix Normalization From the eigenvectors matrix:

I did normalization
but I think there's an error I could not find.
 [/INST] Let$$
v_1=\begin{bmatrix}
1  \\
0 \\
-1
\end{bmatrix}
\quad 
v_2=
\begin{bmatrix}
 1  \\
 \frac{-\rho_2 + \sqrt{((\rho_2)^{2} + 8(\rho_1)^2)}}{2\rho_1} \\
 1 
\end{bmatrix}
\quad
v_3= 
\begin{bmatrix}
 1 \\
 \frac{-\rho_2 - \sqrt{((\rho_2)^{2} + 8(\rho_1)^2)}}{2\rho_1} \\\\
 1
\end{bmatrix}
$$
the column vectors of matrix $\Phi^\ast$.  Note that $\|v_1\|=\sqrt{2}$ and 
\begin{align}
\|v_2\|
=&
\sqrt{
1^2+\left( \frac{-\rho_2 + \sqrt{((\rho_2)^{2} + 8(\rho_1)^2)}}{2\rho_1} \right)^2+1^2}
\\
=&
\sqrt{
2+\frac{\rho_2^2-2\cdot \rho_2\cdot \sqrt{((\rho_2)^{2} + 8(\rho_1)^2)} + ((\rho_2)^{2} + 8(\rho_1)^2)}{4\rho_1^2}}
\\
=&
\sqrt{
\frac{8\rho_1^2+ \rho_2^2-2\cdot \rho_2\cdot \sqrt{((\rho_2)^{2} + 8(\rho_1)^2)} + ((\rho_2)^{2} + 8(\rho_1)^2)}{4\rho_1^2}}
\\
=&
\sqrt{
\frac{\big(16\rho_1^2+ 2\rho_2^2\big)-2\cdot \rho_2\cdot \sqrt{((\rho_2)^{2} + 8(\rho_1)^2)}  }{4\rho_1^2}}
\\
=&
\sqrt{
\frac{\big(16\rho_1^2+ 2\rho_2^2\big)-2\cdot \rho_2\cdot \sqrt{((\rho_2)^{2} + 8(\rho_1)^2)}}{4\rho_1^2}}
\\
=&
\frac{\sqrt{16\rho_1^2+ 2\rho_2^2-2\cdot \rho_2\cdot S}}{2\rho_1}
\end{align}
Similarly, we have
$$
\|v_3\|= \frac{\sqrt{16\rho_1^2+ 2\rho_2^2+2\cdot \rho_2\cdot S}}{2\rho_1}
$$
</s>