The following was taken from Toro's Riemann Solvers and Numerical Methods for Fluid Dynamics, Equation (2.123) in Chapter 2.
It states that for a general quasi-linear hyperbolic system $U_t + A(U)U_x = 0$ with $U = [u_1, u_2, ..., u_m]^T$, where $\lambda_i$ and $R_i = [r_1^{(i)}, r_2^{(i)}, ..., r_m^{(i)}]^T$ correspond to the $i^{th}$ eigenvalue and right eigenvector respectively, the following relations hold true across the $i^{th}$ wave structure:
$$\frac{du_1}{r_1^{(i)}} = \frac{du_2}{r_2^{(i)}} = ... = \frac{du_m}{r_m^{(i)}}$$
Of note, the Riemann invariants along the $i^{th}$ characteristic can be obtained using the left eigenvectors of the Jacobian matrix $A(U)$, shown below for a general quasi-linear hyperbolic system.
$L_iA=\lambda_iL_i$ where $L_i$ is the $i^{th}$ left eigenvector.
Consider the following,
$L_iU_t + \lambda_iL_iU_x$
$=L_i(U_t+\lambda_iU_x)$
$=L_i(-AU_x+\lambda_iU_x)$
$=(-L_iA+\lambda_iL_i)U_x = 0$
since $L_iA=\lambda_iL_i$
$\therefore L_iU_t + \lambda_iL_iU_x = 0$
Let $L_i^T = \nabla{Q_i}$ where $\nabla(*) = [\partial{(*)}/\partial{u_1}, ..., \partial{(*)}/\partial{u_m}]$ and $Q_i$ is some scalar function,
$\nabla{Q_i} \cdot U_t + \lambda_i\nabla{Q_i} \cdot U_x = 0$
Expanding the dot products and applying chain rule,
$$\frac{\partial{Q_i}}{\partial{t}} + \lambda_i\frac{\partial{Q_i}}{\partial{x}} = 0$$
$$\frac{dQ_i}{dt} = 0 \text{ along } \frac{dx}{dt} = \lambda_i$$
Hence, by using $L_i^T = \nabla{Q_i}$, one is able to obtain $Q_i$ for the Riemann invariants along the $i^{th}$ characteristic. However, how do I derive the Riemann invariant relations involving the right eigenvectors across the $i^{th}$ characteristic?