Interesting Linear Algebra Theorems

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).