Use of diagonal matrices in vibration analysis?

Because the eigenvalues and eigenvectors correspond to normal modes.

 

Because resonance is an eigenstate.

 

Be careful about that example: https://www.physicsforums.com/threads/expunging-myths-from-the-classrooom.849853

 

Thank you! And 'classroom' was exactly where I got it from ...

EDIT: “My experience is that excitement and interest in physics and engineering is generated not only by what is understood but also by what is not.” - Bernard Feldman, a physics professor at the University of Missouri, wrote in 2006.
A wise man!