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