- #1
CMJ96
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Homework Statement
I want to decompose the following matrix into a product of two level matrices ##V_i##
$$ \begin{bmatrix}
0 & 0 & 1 & 0 \\
0 & \frac{-\sqrt{3}}{2} & 0 & \frac{-1}{2} \\
\frac{\sqrt{3}}{2} & \frac{-1}{4} & 0 & \frac{\sqrt{3}}{4} \\
\frac{1}{2} & \frac{\sqrt{3}}{4} & 0 & \frac{-3}{4}
\end{bmatrix} $$
Homework Equations
I have only been given the 3x3 case, which I would like to extend to 4x4, in the 3x3 case the decomposition looks like
$$ U_3 U_2 U_1 U = I_n $$
Where ## I_n ## is the identity matrix.
$$ U= V_1 V_2 V_3 $$
Where ##V_i = U_i ^{\dagger} ##
The Attempt at a Solution
I need to eliminate each entry below the ## u_{i=j} ## terms (if that makes any sense).
Since ## u_{2,1} ## is ##0## I can set ##U_1 = I_n##, for ##u_{3,1}## I start to run into trouble, I know that for a 3x3 matrix I can eliminate this term by setting ##U_2## to
$$ \begin{bmatrix}
\frac{u_{1,1}^*}{n} & 0 & \frac{u_{3,1}^*}{n} \\
0 & 1 & 0 \\
\frac{u_{3,1}^*}{n}& 0 & -\frac{u_{1,1}^*}{n}
\end{bmatrix} $$
Where ##n=\sqrt{u_{1,1}^2 + u_{3,1}^2 } ##
I have attempted to expand this to a 4x4 matrix, and this is what I got
$$ \begin{bmatrix}
\frac{u_{1,1}}{n} & 0 & 0 & \frac{u_{4,1}}{n}\\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
\frac{u_{4,1}}{n} & 0 & 0 & -\frac{u_{1,1}}{n}
\end{bmatrix} $$
Where ##n= \sqrt{u_{1,1}^2+ u_{4,1}^2 } ##
Is this along the right lines?