How do you determine that the characteristic polynomial is the minimal polynomial?

Solution 1:

The characteristic and the minimal polynomial of a matrix

Let $A$ be an $n \times n$ matrix. We associate two polynomials to $A$:

1. The characteristic polynomial of A can defined as $F(X) = \det(X ·I −A)$, where $X$ is the variable of the polynomial, and $I$ represents the identity matrix. $F(X)$ is a monic polynomial of degree $n$.

2. The minimal polynomial of $A$ which we will denote by $\mu(X)$, is defined by the following properties:

• $\mu(X)$ is monic (It means the leading coefficient is $1$),
• $\mu(A)$ = 0,
• $\mu(X)$ is the monic polynomial of the smallest possible degree such that $\mu(A) = 0$
• If $g(X)$ is another polynomial, then $g(A) = 0$ if and only if $\mu(X)$ divides $g(X)$.
• $F(X)$ is a multiple of $\mu(X)$