Summary: This paper is concerned with symmetrization and diagonalization of real matrices and their implications for the dynamics of linear second-order systems governed by equations of motion having asymmetric coefficient matrices. Results in the light of Taussky's theorem [\textit{O. Taussky}, Adv. Math. 2, 175-186 (1968; Zbl 0197.02702)] are presented. The connection of the symmetrizers with the eigenvalue problem is brought out. An alternative proof of Taussky's theorem for real matrices is presented. Diagonalization of two real symmetric (but not necessarily positive-definite) matrices is discussed in the context of undamped non-gyroscopic systems. A commutator of two matrices with respect to a given third matrix is defined; this commutator is found to play an interesting role in deciding simultaneous diagonalizability of two or three matrices. Errors in a few previously known results are brought out. Pseudo-conservative systems are studied, and their connection with the so-called `symmetrizable systems' is critically examined. Results for modal analysis of general non-conservative systems are presented. Illustrative examples are given.