The general solution of the matrix equation $w_t+\sum\limits_{k=1}^nw_{x_k}\rho^{(k)}(w)=\rho(w)+[w,T\tilde\rho(w)]$

We construct the general solution of the equation $w_{t} + \sum_{k = 1}^{n} w_{x_{k}} ρ^{(k)} (w) = ρ (w) + [w, T \tilde{ρ} (w)]$ , for the $N \times N$ matrix $w$ , where $T$ is any constant diagonal matrix, $n, N \in \NN_+ $a n d$ \rho^{(k)}, \rho, \tilde\rho: \RR \to \RR$ are arbitrary analytic functions. Such a solution is based on the observation that, as $w$ evolves according to the above equation, the evolution of its spectrum decouples, and it is ruled by the scalar analogue of the above equation. Therefore the eigenvalues of $w$ and suitably normalized eigenvectors are the $N^{2}$ Riemann invariants. We also obtain, in the case $ρ = \tilde{ρ} = 0$ , a system of $N^{2}$ non-differential equations characterizing such a general solution. We finally discuss reductions of the above matrix equation to systems of $N$ equations admitting, as Riemann invariants, the eigenvalues of $w$ . The simplest example of such reductions is a particular case of the gas dynamics equations

The general solution of the matrix equation $w_{t} + \sum_{k = 1}^{n} w_{x_{k}} ρ^{(k)} (w) = ρ (w) + [w, T \tilde{ρ} (w)]$

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The general solution of the matrix equation $w_{t} + \sum_{k = 1}^{n} w_{x_{k}} ρ^{(k)} (w) = ρ (w) + [w, T \tilde{ρ} (w)]$

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The general solution of the matrix equation wt+∑k=1nwxkρ(k)(w)=ρ(w)+[w,Tρ~(w)]wt​+k=1∑n​wxk​​ρ(k)(w)=ρ(w)+[w,Tρ~​(w)]

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The general solution of the matrix equation wt+∑k=1nwxkρ(k)(w)=ρ(w)+[w,Tρ~(w)]wt​+k=1∑n​wxk​​ρ(k)(w)=ρ(w)+[w,Tρ~​(w)]

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The general solution of the matrix equation $w_{t} + \sum_{k = 1}^{n} w_{x_{k}} ρ^{(k)} (w) = ρ (w) + [w, T \tilde{ρ} (w)]$

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The general solution of the matrix equation $w_{t} + \sum_{k = 1}^{n} w_{x_{k}} ρ^{(k)} (w) = ρ (w) + [w, T \tilde{ρ} (w)]$

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