Matrix representation of the $x$-component of orbital angular momentum $hat{L}_x$

Question

In my notes it is given that using the spherical harmonic (shown below) as basis states in this order, the matrix representing the $x$-component of orbital angular momentum $hat{L}_x$ for a particle with angular momentum quantum number $l=1$:
$$
qquad
hat L_x=frac{hbar}{sqrt{2}} left(
begin{array}{ccc}
 0 & 1 & 0 
 1 & 0 & 1 
 0 & 1 & 0 
end{array}
right) tag{1}
$$
The spherical harmonic used are:

I am trying to derive the matrix of $hat{L}_x$.
First I need to find the eigenvlaues and eigenvectors of $hat{L}_x$, however I cannot find it anywhere and I don't know how to derive it.

MiguelFuego · Answer

You can derive $L_x$ from the ladder operators $L_{pm}$.
These operators verify:
begin{equation}
L_{pm} | 1, m rangle= hbar sqrt{2 - m (m pm 1)} | 1, m pm 1 rangle
end{equation}
With the given basis you can derive the matrix for these operators and find $L_x$ like:
begin{equation}
L_x = frac{1}{2} (L_+ + L_-)
end{equation}

Matrix representation of the $x$-component of orbital angular momentum $hat{L}_x$

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