theorems – Mathtuition88

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

Tag: theorems

Interesting Linear Algebra Theorems

Diagonalizable & Minimal Polynomial

Characteristic Polynomial

Cayley-Hamilton Theorem