How to construct all Hermitian and unitary 3x3 matrices?

Question

2x2 matrices of Pauli are $\sigma_x$, $\sigma_y$, and $\sigma_z$, with

$$\sigma_x = \pmatrix{0 && 1 \\ 1 &&0},~\sigma_y = \pmatrix{0 && -i \\ i &&0}, \sigma_z = \pmatrix{1 && 0 \\ 0 && -1}$$

are Hermitian and unitary.

Question: How to construct all Hermitian and unitary 3x3 matrices?

3x3 matrices of Gell-Mann are $\lambda_1$, $\lambda_2$, $\lambda_3$, $\lambda_4$, $\lambda_5$, $\lambda_6$, $\lambda_7$, $\lambda_8$, with example

$$\lambda_1= \pmatrix{0 && 1 && 0 \\ 1 &&0 &&0 \\ 0 && 0 &&0}$$

But they cannot be inverted and so are not unitary.

Answer 1

By spectral theory, a matrix is Hermitian iff it has an orthonormal basis of eigenvectors with real eigenvalues, and a matrix is unitary iff it has an orthonormal basis of eigenvectors with eigenvalues of absolute value $1$. SCombining these conditions, a matrix is both Hermitian and unitary iff it has an orthonormal basis of eigenvectors with eigenvalues $\pm1$.

So, you can construct Hermitian and unitary $3\times 3$ matrices by first choosing some orthonormal basis $\{e_1,e_2,e_3\}$ and then taking the matrix $A$ such that $Ae_1=\lambda_1e_1$, $Ae_2=\lambda_2e_2$, and $Ae_3=\lambda_3e_3$ where each $\lambda_i$ is $\pm1$. Or in other words, you can take a matrix of the form $\begin{pmatrix} \pm 1 & 0 & 0 \\ 0 & \pm1 & 0 \\ 0 & 0 & \pm1 \end{pmatrix}$ and conjugate it by any unitary matrix.