# 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.

## Alumni Liaison

Basic linear algebra uncovers and clarifies very important geometry and algebra.

Dr. Paul Garrett