## Diagonalizable & Minimal Polynomial

A matrix or linear map is diagonalizable over the field $F$ if and only if its minimal polynomial is a product of distinct linear factors over $F$.

## Characteristic Polynomial

Let $A$ be an $n\times n$ matrix. The characteristic polynomial of $A$, denoted by $p_A(t)$, is the polynomial defined by $\displaystyle p_A(t)=\det(tI-A).$

## Cayley-Hamilton Theorem

Every square matrix over a commutative ring satisfies its own characteristic equation:

If $A$ is an $n\times n$ matrix, $p(A)=0$ where $p(\lambda)=\det(\lambda I_n-A)$.