# The Importance of Real Orthonormal Eigenvectors in Classically Damped Systems

The equations of motion of an $$n$$-degree-of-freedom linear dynamic system with viscous damping may be written in the form \begin{align} M \ddot x + C \dot x + K x = 0. \end{align} This system has been studied in the literature. If this system is classically damped, then the system has a complete set of real orthonormal eigenvectors namely the classical normal modes. I was wondering why having real eigenvectors is important. My major is not mechanical engineering, so I greatly appreciate any comment/insight. Thank you.

• This is a trivial equation studied in elementary courses. What do you mean by "comment/insight"? – John B Jan 25 at 18:53
• For example, what are the consequences if $M$, $C$, and $K$ are such that the system does not have real eigenvectors? Why do we care about having real eigenvector? I also greatly appreciate it if you could introduce some references (as you said in elementary course) that rigorously covers this topic. Thanks. – Arthur Jan 26 at 2:31
• Just pick up your favorite book. Something such as springer.com/gp/book/9781468401738 should be of help. – John B Jan 26 at 6:22
• Thanks for sharing this reference. I looked into the second chapter "Second-order linear differential equations". It seems that it is discussing the scalar equations (rather than matrix equations), and does not answer my question about the role of eigenvectors. The book mainly talks about different ways for solving this equation. – Arthur Jan 26 at 16:56
• Then just pass to a scalar equation on a higher-dimensional space. Nothing changes. You really need to study the theory. – John B Jan 27 at 8:25