Uniformly symmetrizable \(3\times 3\) matrices (Q1826813)

The Cauchy problem for a hyperbolic system with constant coefficients of the form \(u_t= \sum^n_{j=1} A_j u_{x_j}+ Bu\) is well-posed in \(C^\infty\) for any \(B\), if and only if the family of matrices \(\Sigma_j A_j\xi_j\) is uniformly symmetrizable for \(\xi_1,\dots,\xi_n\) running in the real line \(\mathbb{R}\). \textit{H.-O. Kreiss} [Math. Scand. 7, 71--80 (1959; Zbl 0090.09801)] gave the following characterization of the uniform symmetrizability (US): \[ \{A(t)\}\text{ is US } \Leftrightarrow\|(A(t)- zI)^{-1}\|\leq {C\over\text{Im\,}z}\text{ for }\text{Im\,}z> 0. \] The purpose of this paper is to prove a more intrinsic characterization, at least for matrices of order 2 or 3. Section 2 contains five prelimary remarks. Sections 3 and 4 are proofs of necessary and sufficient conditions for the uniform symmetrizability of a family of matrices. Sections 5 and 6 are for the cases of triangular and \(2\times 2\) matrices.