The authors study the Cauchy problem for systems of linear differential equations with constant coefficients \[ {\partial U\over\partial t}= P\Biggl({\partial\over\partial x}\Biggr) U,\quad t>0,\quad U(x, 0)= U_0(x), \] where \(U\) is a vector function of \(x\), \(t\). They prove two new criteria for the sufficiency of the Neumann condition for the stability of difference schemes for such systems. The first criterion is that the von Neumann criterion is sufficient for stability if a finite power of the amplification matrix is a uniformly diagonalizable matrix. The second criterion relaxes the uniform diagonalizability for the amplification matrix. The authors investigate the satisfaction of the obtained uniform stability criteria for a number of well-known difference schemes for the numerical solution of fluid dynamics problems. The paper is carefully written and contains both a thorough mathematical investigation and numerical tests.