Eigenvector

These are vectors that remain in their same span after some Linear Transform.

Motivation

We want to find an optimal basis to use for our Linear Map, so that computations are easier.

Definition

• With linear map $T:V\to V$
• If $T(x)=\lambda x$ for a non-zero $x\in V$
• $x$ is the eigenvector
• $\lambda$ is the eigenvalue

Concepts

• Finding Eigenvectors
• Eigenvalue Midpoint Trick
• LADR Finding Eigenvectors to Find a Dependence
• LADR Factor and Apply the Dependence
• Algebraic Multiplicity
• Geometric Multiplicity
• Geometry ⇐ Algebra Theorem

Theorems

• Determinants and Eigenvalues
• Invertibility and Eigenvalues