# Eigen_Value

An Eigen Value is the constant, $ \lambda $ by which an Eigen Vector, $ \vec v $ is multiplied to satisfy the equation

$ A\vec v = \lambda \vec v $

## Determination

Finding the Eigen Value is actually quite simple. First you move all the terms in the defining equation to the right hand side, then group terms.

$ 0=(\lambda I_n-A)\vec v $

Where $ I_n $ is the $ n^{th} $ identity matrix

Since $ \vec v $ must be a non-trivial (non-zero) vector, the only way this solution can happen is if the determinant of the effective matrix is zero.

So we now have

$ det(\lambda I_n-A)=0 $

Where $ det() $ is the determinant. When the left hand side of this equation is computed, a characteristic polynomial with order n will result with respect to the Eigen Value, $ \lambda $. Solving this equation polynomial for its zeros gives the Eigen Values. The number of times an Eigen Value appears as a zero to the characteristic polynomial is called the algebraic multiplicity of that Eigen Value.

These Eigen Values found can now be plugged back in to find the corresponding Eigen Vectors.