# Riemann Problem for Linear Hyperbolic Systems

In this post, I will be solving the linear system

with the Riemann initial conditions

where and is an n by n matrix which has distinct real eigenvalues.

Since has real distinct eigenvalues, we know that the matrix can be decomposed as

where columns in correspond to ‘s eigenvectors and is a diagonal matrix containing the eigenvalues

Additionally, because we have distinct real eigenvalues, we know that is a unitary matrix ().

Because is unitary, we can uniquely decompose a vector into

Using this fact, it is beneficial to write and as

where and .

Now, we will transform the original pde using to

Likewise, we can rewrite the initial conditions as

where is a unit vector.

We can conclude that the original system decouples to n linear advection equations

which has the solution

Therefore, we can solve the original equation by using the transformation which gives us