Some Linear Algebra Theorems

Linear Algebra

Diagonalizable & Minimal Polynomial:
A matrix or linear map is diagonalizable over the field if and only if its minimal polynomial is a product of distinct linear factors over .

Characteristic Polynomial:
Let be an matrix. The characteristic polynomial of , denoted by , is the polynomial defined by

Cayley-Hamilton Theorem:
Every square matrix over a commutative ring satisfies its own characteristic equation:

If is an matrix, where .