Eigenstuff

Eigenvectors and Eigenvalues

Given

1. A is square
2. defined, e.g. if then

Eigenvector

is an eigenvector for An eigenvector is a vector solution to the above equation, such that the linear transformation of has the same result as scaling the vector by .

Eigenvalue

is the corresponding eigenvalue () Solve for in , which yields the Characteristic Equation for this system, e.g. in a 2D systems it is . In a 3D+ system, you still have to create the characteristic equation but it requires

Notes:

• point same direction
• point opposite direction
• 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 are eigenvectors that correspond to distinct eigenvalues, then are linearly independent

Defective

An eigenvalue is defective if and only if it does not have a complete Set of Linearly Independent eigenvectors.
Due to ‘s contribution,

Neutral Eigenvalue

Eigenspace

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