Cayley Hamilton Theorem

A theorem to state that the Characteristic Polynomial is a T-Annihilator. Used to find the inverse of matrixes

Theorem

• Let $T\in \mathbb{L}(V)$
• If $dim(V)<\infty$
• $f(X)=\det (xI-T)$ is the Characteristic Polynomial for $T$
• Then, $f(T)=0$

Corrolaries

• Minimal Polynomial Divides Characteristic Polynomial Theorem
• Minimal Polynomial has Same Roots as Characteristic Polynomial
• Minimal Polynomial has Same Prime Factors as Characteristic Polynomial

Examples

• Cayley Hamilton Theorem to Find Inverse