recently in my researches I've come across the following operator


$$L\left(\begin{array}{c}
 a\_1\\
 \vdots\\
 a\_n
 \end{array}\right)=M\_1\left(\begin{array}{c}
 \partial\_{z\_1}a\_1\\
 \vdots\\
 \partial\_{z\_1}a\_n
 \end{array}\right)+\dotso+M\_n\left(\begin{array}{c}
 \partial\_{z\_n}a\_1\\
 \vdots\\
 \partial\_{z\_n}a\_n
 \end{array}\right),$$
where $a\_1,\dotso,a\_n:\mathbb R^n\to\mathbb R$ are smooth functions of the independent variables $z\_1,\dotso,z\_n$ and $M\_1,\dotso M\_n$ are real square $n\times n$ matrices with constant coefficients (though none of them is invertible).


Is there a general theory to treat this simple system of linear first order pdes? Is it possible to write explicitly a solution if we were to solve


$$L\left(\begin{array}{c}
 a\_1\\
 \vdots\\
 a\_n
 \end{array}\right)=\left(\begin{array}{c}
 b\_1\\
 \vdots\\
 b\_n
 \end{array}\right),$$
for some smooth functions $b\_1,\dotso,b\_n:\mathbb R^n\to\mathbb R$?


Thank you for the references and the patience.


Kind regards,


Guido


**Edit**


Following the suggestions, I report here the $3\times 3$ system which is the toy model I'm currently studying


$$\left(\begin{array}{ccc}
 B\_{11} & B\_{12} & 0\\
 -B\_{11} & 0 & 0\\
 0 & -B\_{12} & 0
 \end{array}
 \right)\left(\begin{array}{c}
\partial\_{z\_1}a\_1\\
\partial\_{z\_1}a\_2\\
\partial\_{z\_1}a\_3\\
 \end{array}\right)+
\left(\begin{array}{ccc}
 0 & 0 & -B\_{23}\\
 0 & B\_{22} & B\_{23}\\
 0 & -B\_{22} & 0
 \end{array}
 \right)\left(\begin{array}{c}
\partial\_{z\_2}a\_1\\
\partial\_{z\_2}a\_2\\
\partial\_{z\_2}a\_3\\
 \end{array}\right)+
\left(\begin{array}{ccc}
 0 & 0 & -B\_{33}\\
 -B\_{31} & 0 & 0\\
 B\_{31} & 0 & B\_{33}
 \end{array}
 \right)\left(\begin{array}{c}
\partial\_{z\_3}a\_1\\
\partial\_{z\_3}a\_2\\
\partial\_{z\_3}a\_3\\
 \end{array}\right)=
\left(\begin{array}{c}
 b\_1\\
 b\_2\\
 -b\_1-b\_2
 \end{array}\right),
$$


$a\_i$ and $b\_i$ are smooth functions, the $b\_i$'s are compactly supported. I'm interested in solving locally the above system. I think it has something to do with the usual theory for the hyperbolic systems of differential equations, however I'm to unexperienced in the field to be sure about the steps to follow. I understand the condition to be imposed on the functions $b\_i$ as a necessary condition for the system to be possibly solvable, but it is not clear to me why there should exist a solution at all! This is the most I can say up to now. Thanks again for the patience and the help



System of linear first order PDE with constant coefficients
Since you wrote the system explicitly, here's an explicit solution using Fourier transforms. Let $a(z) = \int e^{ik\cdot z} a(k)\, d^3k $ and $b(z) = \int e^{ik\cdot z} b(k)\, d^3k$. The $a(k)$ and $b(k)$ vectors will satisfy the Fourier transformed equation $iM(k) a(k) = b(k)$, where $M(k) = (M\_1 k\_1 + M\_2 k\_2 + M\_3 k\_3)$. It can be checked explicitly that the following formula gives a complete solution for $a(k)$:
$$\begin{gathered}
\begin{pmatrix} a\_1(k) \\ a\_2(k) \\ a\_3(k) \end{pmatrix} =
\frac{i}{\Delta(k)}
\begin{pmatrix}
 B\_{23} k\_2 & B\_{23} k\_2 + B\_{33} k\_3 \\
 0 & 0 \\
 B\_{11} k\_1 + B\_{31} k\_3 & B\_{11} k\_1
\end{pmatrix}
\begin{pmatrix} b\_1(k) \\ b\_2(k) \end{pmatrix}
+ \begin{pmatrix} \Delta\_1(k) \\ \Delta\_2(k) \\ \Delta\_3(k) \end{pmatrix} c(k), \\
\Delta(k) = B\_{11} B\_{33} k\_1 k\_3 + B\_{23} B\_{31} k\_2 k\_3 + B\_{31} B\_{33} k\_3^2 , \\
\Delta\_1(k) = B\_{12} B\_{23} k\_1 k\_2 + B\_{22} B\_{23} k\_2^2 + B\_{22} B\_{33} k\_2 k\_3 , \\
\Delta\_2(k) = B\_{11} B\_{33} k\_1 k\_3 + B\_{23} B\_{31} k\_2 k\_3 + B\_{31} B\_{33} k\_3^2 , \\
\Delta\_3(k) = B\_{11} B\_{12} k\_1^2 + B\_{11} B\_{22} k\_1 k\_2 + B\_{12} B\_{31} k\_1 k\_3 ,
\end{gathered}
$$
where $c(k)$ is an arbitrary scalar function.


Of course, because the denominator $\Delta(k)$ vanishes for some $k$, the Fourier coefficients $a(k)$ must be interpreted as distributions.


