Eigenstuff

Eigenvectors and Eigenvalues

Given

1. A is square
2. $A\vec{v}$ defined, e.g. if $A\in {\mathbb{R}}^{n\times n}$ then $\vec{v}\in {\mathbb{R}}^n$

$$A\vec{v}=\lambda \vec{v}$$

Eigenvector

$\vec{v}$ is an eigenvector for $A$ An eigenvector is a vector solution $\vec{v}$ to the above equation, such that the linear transformation of $A$ has the same result as scaling the vector by $\lambda$.

Eigenvalue

$\lambda$ is the corresponding eigenvalue ($\lambda \in \mathbb{C}$) $A\vec{v}=\lambda \vec{v}⟹\det (A-\lambda I)=0$ Solve for $\lambda$ in $\det (A-\lambda I)=0$, which yields the Characteristic Equation for this system, e.g. in a 2D systems it is ${\lambda }^2-\text{tr}A\cdot \lambda +\det A=0$. In a 3D+ system, you still have to create the characteristic equation but it requires

Notes:

• $\lambda >0⟹A\vec{v},\vec{v}$ point same direction
• $\lambda <0⟹A\vec{v},\vec{v}$ point opposite direction
• $\lambda$ can be complex even if nothing else in the equation is
• Eigenvalues cannot be determined from the reduced version of a matrix ⭐
  • i.e. row reductions change the eigenvalues of a matrix
• The diagonal elements of a triangular matrix are its eigenvalues.
• A invertible iff 0 is not an eigenvalue of A.
• Stochastic matrices have an eigenvalue equal to 1.
• If ${\vec{v}}_1,{\vec{v}}_2,...,{\vec{v}}_k$ are eigenvectors that correspond to distinct eigenvalues, then ${\vec{v}}_1,{\vec{v}}_2,...,{\vec{v}}_k$ are linearly independent

Defective

An eigenvalue is defective if and only if it does not have a complete Set of Linearly Independent eigenvectors.
$\lambda =0⟹\text{ Defective}$ Due to $\mathrm{Im}\lambda$‘s contribution, $\mathrm{Re}\lambda =0/⟹\text{ Defective}$

Neutral Eigenvalue

$$\forall \lambda s.t.\mathrm{Re}\lambda =0$$

Eigenspace

• the span of the eigenvectors that correspond to a particular eigenvalue

• $Nul(A-\lambda I)$