Anyone able to derive the following Riemann invariant relations? 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?
 A: Eq. (2.123) of Toro is a definition of the so-called Generalised Riemann Invariants. I will briefly present the theory by using the notations in OP. I am using the book Numerical Approximation of Hyperbolic Systems of Conservation Laws by Godlewski and Raviart (Springer, 1996), Section 3.2 p. 53.
Definition. A smooth scalar function $U \mapsto Q_i(U)$ is called a $i$-Riemann invariant if it satisfies $ \nabla Q_i \cdot R_i = 0$ for all $U$ in ${\bf R}^m$, where $R_i$ is the $i$th right eigenvector.
Now, if we write the Lagrange-Charpit equations for $\nabla Q_i \cdot R_i = 0$, we find
$$
\frac{dU}{d\xi} = R_i
$$
where we have introduced the parametrization $U = U(\xi)$ to write $\frac{d}{d\xi} Q_i = \nabla Q_i \cdot \frac{d}{d\xi} U$. The proposed identity
$$
\frac{d u_1}{r_1^{(i)}} = \dots = \frac{d u_m}{r_m^{(i)}}
$$
is only a parameter-invariant form of the latter equation.
Note: if we insert the above parametrization in the PDE system $U_t + A(U)U_x = 0$, then we find
$$
\left(\xi_t I + \xi_x A(U)\right) \frac{d U}{d\xi} = \left(\xi_t + \lambda_i \xi_x\right) R_i = 0
$$
where we have used $\frac{d}{d\xi} U = R_i$ and the definition of the eigenvector. Thus, we arrive at the scalar transport equation $\xi_t + \lambda_i \xi_x = 0$ for the parameter $\xi$, and the $i$-Riemann invariant satisfies the PDE $(\partial_t + \lambda_i\partial_x)Q_{i} = 0$ proposed in OP.
