Eigenvalue & Eigenvector

Definition

\[A\mathbf{v} = \lambda \mathbf{v}\]

• \(\lambda\): eigenvalue
• \(\mathbf{v} \neq \mathbf{0}\): eigenvector

Characteristic Polynomial

\[p(\lambda) = \det(A - \lambda I)\]

Roots = eigenvalues (including complex).

Diagonalization

\[A = PDP^{-1}\]

• \(D\): diagonal of eigenvalues
• \(P\): matrix of eigenvectors

Condition: \(n\) linearly independent eigenvectors.

Spectral Theorem

\[A = Q\Lambda Q^T\]

• \(Q\): orthogonal matrix of eigenvectors
• \(\Lambda\): diagonal eigenvalue matrix

Cayley-Hamilton

\[p_A(A) = 0\]

Matrix satisfies its own characteristic polynomial.

Minimal Polynomial

Smallest degree monic \(m(\lambda)\) with \(m(A)=0\).

Divides characteristic polynomial, shares roots.