Given a system of n conservation equations for the n conserved variables in m-dimensional space:
Note: the choice of conserved variables is unique. The conservation equations can be transformed into ``primitive'', wave-like equations for a corresponding set of n non-unique primitive variables ,
where the are matrices. The choice of primitive variables is not unique, and could be taken to be the conserved variables. Assume the two systems are related by
Then the conservation equations can be transformed as follows:
The eigenvalues of are the solution of
The system is hyperbolic if all the eigenvalues of all the are real. The left (row), , and right (column), , eigenvectors of , satisfy
The left and right eigenvectors are orthogonal,
The matrices can be individually, but not simultaneously, diagonalized by a similarity transformation
where the columns of are the right eigenvectors and rows of its inverse are the left eigenvectors , and is the diagonal matrix, .
The Characteristic Equations for a single direction can be found as follows: Combine all the divergence terms except that for the desired direction, say 1, with the source term, so the conservation equations become
and the primitive equations become
The 1D Characteristic Equations are obtained by multiplying this last equation by
or in component form
which gives the variation of along the characteristic curve in the -plane defined by
tangent to the eigenvector with eigenvalue .
Sometimes it is possible to define a CHARACTERISTIC VARIABLE or Riemann Invariant by
if the expression is a perfect differential (the coefficients satisfy Pfaff's condition for the integrability of differential forms or no more than two differentials appear on the right hand side). Then the characteristic equation (13) becomes
These are wave equations with characteristic speeds and with Riemann Invariant constant along the characteristic .