1. Any equation which has a solution, subject to specified boundary conditions, only when a parameter occurring in it has certain values.

2. Specifically, the equation Au = λu, which can have a solution only when the parameter λ has certain values, where A can be a square matrix which multiplies the vector u, or a linear differential or integral operator which operates on the function u, or in general, any linear operator operating on the vector u in a finite or infinite dimensional vector space. Also known as eigenvalue equation.

3. An equation which sets the characteristic polynomial of a given linear transformation on a finite dimensional vector space, or of its matrix representation, equal to zero.

An equation relating a set of variables, such as pressure, volume, and temperature, whose values determine a substance's physical condition.

An equation whose solutions give the frequencies and modes of those perturbations of a hydromagnetic system which decay or grow exponentially in time, and indicate regions of stability of such a system.