On Analytical Families of Matrices Generating Bounded Semigroups

We consider linear difference schemes with several degrees of freedom (DOFs) per cell for a transport equation with a constant coefficient. The Fourier transform decomposes the scheme into a number of finite systems of ODEs, the number of equations in each system being equal to the number of DOFs. The matrix of these systems is an analytical function of the wave vector. Generally this matrix is not diagonalizable and, if it is, the diagonal form may be non-smooth. We show that for L2-stable schemes in the 1D case the matrix can be locally transformed to a block-diagonal form preserving the analytical dependence on the wave number.