1. Identify the Quantum State Basis:

   Use the eigenstates of the \( L^2 \) and \( L_z \) operators. For a system with total angular momentum quantum number \( l \), the states are labeled as \( |l, m\rangle \), where \( m \) ranges from \(-l\) to \(l\).

2. Matrix Elements of \( L_z \):

   The operator \( L_z \) is diagonal in the \( |l, m\rangle \) basis:

   \[ \langle l, m' | L_z | l, m \rangle = \hbar m \delta_{m', m} \]

   Thus, the matrix representation of \( L_z \) is diagonal, with entries \( \hbar m \).

3. Matrix Elements of \( L^2 \):

   Similarly, \( L^2 \) is also diagonal:

   \[ \langle l, m' | L^2 | l, m \rangle = \hbar^2 l(l + 1) \delta_{m', m} \]

   The matrix will have the same structure, with the eigenvalue \( \hbar^2 l(l + 1) \).

4. Matrix Elements of \( L_x \) and \( L_y \):

   These operators can be expressed in terms of the raising and lowering operators \( L_+ \) and \( L_- \):

   \[ L_+ = \hbar \sqrt{(l - m)(l + m + 1)} |l, m + 1\rangle \langle l, m| \]
   \[ L_- = \hbar \sqrt{(l + m)(l - m + 1)} |l, m - 1\rangle \langle l, m| \]

   The relations for \( L_x \) and \( L_y \) are:

   \[ L_x = \frac{1}{2}(L_+ + L_-) \]
   \[ L_y = \frac{1}{2i}(L_+ - L_-) \]
5. Constructing the Matrices:

   For a given \( l \), create the \( (2l + 1) \times (2l + 1) \) matrices for \( L_x \) and \( L_y \) by calculating the appropriate matrix elements using the above definitions. Each matrix will be filled based on the transition probabilities defined by the raising and lowering operators.

6. Example for \( l = 1 \):

   For \( l = 1 \), the basis states are \( |1, 1\rangle, |1, 0\rangle, |1, -1\rangle \).

   Construct \( L_z \) and \( L^2 \) as diagonal matrices:

   \[ L_z = \hbar \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}, \quad L^2 = \hbar^2 2 \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

   Calculate \( L_+ \) and \( L_- \) using the raising and lowering operators.

For \( l = \frac{1}{2} \)

Basis States: \( | \frac{1}{2}, \frac{1}{2} \rangle, | \frac{1}{2}, -\frac{1}{2} \rangle \)

Matrix Forms:

1. \( L_z \)
   \[ L_z = \hbar \begin{pmatrix} \frac{1}{2} & 0 \\ 0 & -\frac{1}{2} \end{pmatrix} \]
2. \( L^2 \)
   \[ L^2 = \hbar^2 \frac{3}{4} \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]
3. \( L_+ \)
   \[ L_+ = \hbar \begin{pmatrix} 0 & \sqrt{1} \\ 0 & 0 \end{pmatrix} \]
4. \( L_- \)
   \[ L_- = \hbar \begin{pmatrix} 0 & 0 \\ \sqrt{1} & 0 \end{pmatrix} \]
5. \( L_x \) and \( L_y \)

   Using \( L_x = \frac{1}{2}(L_+ + L_-) \) and \( L_y = \frac{1}{2i}(L_+ - L_-) \):

   \[ L_x = \frac{\hbar}{2} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \]
   \[ L_y = \frac{\hbar}{2} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} \]

For \( l = 1 \)

Basis States: \(|1, 1\rangle, |1, 0\rangle, |1, -1\rangle \)

Matrix Forms:

1. \( L_z \)
   \[ L_z = \hbar \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix} \]
2. \( L^2 \)
   \[ L^2 = \hbar^2 2 \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]
3. \( L_+ \)
   \[ L_+ = \hbar \begin{pmatrix} 0 & \sqrt{2} & 0 \\ 0 & 0 & \sqrt{2} \\ 0 & 0 & 0 \end{pmatrix} \]
4. \( L_- \)
   \[ L_- = \hbar \begin{pmatrix} 0 & 0 & 0 \\ \sqrt{2} & 0 & 0 \\ 0 & \sqrt{2} & 0 \end{pmatrix} \]
5. \( L_x \) and \( L_y \)
   \[ L_x = \frac{\hbar}{2} \begin{pmatrix} 0 & \sqrt{2} & 0 \\ \sqrt{2} & 0 & \sqrt{2} \\ 0 & \sqrt{2} & 0 \end{pmatrix} \]
   \[ L_y = \frac{\hbar}{2} \begin{pmatrix} 0 & -i\sqrt{2} & 0 \\ i\sqrt{2} & 0 & -i\sqrt{2} \\ 0 & i\sqrt{2} & 0 \end{pmatrix} \]

For \( l = \frac{3}{2} \)

Basis States: \( | \frac{3}{2}, \frac{3}{2} \rangle, | \frac{3}{2}, \frac{1}{2} \rangle, | \frac{3}{2}, -\frac{1}{2} \rangle, | \frac{3}{2}, -\frac{3}{2} \rangle \)

Matrix Forms:

1. \( L_z \)
   \[ L_z = \hbar \begin{pmatrix} \frac{3}{2} & 0 & 0 & 0 \\ 0 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & -\frac{1}{2} & 0 \\ 0 & 0 & 0 & -\frac{3}{2} \end{pmatrix} \]
2. \( L^2 \)
   \[ L^2 = \hbar^2 \frac{15}{4} \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]
3. \( L_+ \)
   \[ L_+ = \hbar \begin{pmatrix} 0 & \sqrt{3} & 0 & 0 \\ 0 & 0 & \sqrt{2} & 0 \\ 0 & 0 & 0 & \sqrt{3} \\ 0 & 0 & 0 & 0 \end{pmatrix} \]
4. \( L_- \)
   \[ L_- = \hbar \begin{pmatrix} 0 & 0 & 0 & 0 \\ \sqrt{3} & 0 & 0 & 0 \\ 0 & \sqrt{2} & 0 & 0 \\ 0 & 0 & \sqrt{3} & 0 \end{pmatrix} \]
5. \( L_x \) and \( L_y \)
   \[ L_x = \frac{\hbar}{2} \begin{pmatrix} 0 & \sqrt{3} & 0 & 0 \\ \sqrt{3} & 0 & \sqrt{2} & 0 \\ 0 & \sqrt{2} & 0 & \sqrt{3} \\ 0 & 0 & \sqrt{3} & 0 \end{pmatrix} \]
   \[ L_y = \frac{\hbar}{2} \begin{pmatrix} 0 & -i\sqrt{3} & 0 & 0 \\ i\sqrt{3} & 0 & -i\sqrt{2} & 0 \\ 0 & i\sqrt{2} & 0 & -i\sqrt{3} \\ 0 & 0 & i\sqrt{3} & 0 \end{pmatrix} \]