**1. Introduction**

Every mechanical system has the property of vibration. Analog phenomenon can be observed in the electrical systems in the form of oscillation circuits. Neglecting terms of vibration can lead to resonance, which on one hand can have catastrophic consequences as the demolition of the bridge, on the other hand can be used positively. Mathematical description of vibratory condition leads to a differential equation or a system of differential equations. This problem is further transformed to the eigenvalue problem. This is the motivation for considering the eigenvalue problems.

This chapter is organized as follows: The Linear eigenvalue problem and the quadratic eigenvalue problem.

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