## Some Linear Algebra Theorems

Linear Algebra

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 $p(\lambda)=\det(\lambda I_n-A)$.

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