Pauli matrices as a basis of $2\times 2$ Hermitian matrices

Consider the Pauli matrices, \[ \sigma_1 = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}, \quad \sigma_2 = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}, \quad \sigma_3 = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}. \]

1. Calculate \(\sigma_i^\dagger\), \(i = 1, 2, 3\).

2. Decompose a matrix \(A = \begin{bmatrix} a & b + ic \\ b - ic & -a \end{bmatrix}\), \(a, b, c \in \mathbb{R}\), into a sum of \(\sigma_i\).

3. Prove that \(\sigma_i\), \(i = 1, 2, 3\), constitute a basis in the space of \(2 \times 2\) hermitian matrices with zero trace.

4. Prove that \(\{\sigma_i, i = 1, 2, 3;\ 1\}\) constitute a basis in the space of \(2 \times 2\) hermitian matrices.

5. Calculate the eigenvalues and the eigenvectors of \(\sigma_i\), \(i = 1, 2, 3\).