notes on matrix polynomials (minimal polynomial)

A matrix is nonderogatory if every eigenvalue has geometric multiplicity 1.

Theorem: A is nonderogatory iff that commutes with A is a polynomial in A. (Claim: If S^{-1} B S = f(J), then B = f(A).) (The inverse is not proved on textbook.)

Theorem: Every matrix A can be written $A = A_D + A_N$, where $A_D$ diagonalizable, $A_N$ nilpotent, and $A_D$ and $A_N$ commute.

Corollary: $A^m \to 0$, as $m \to \infty$ iff $|\lambda| < 1$ for any eigenvalue of A.

An annihilatory polynomial is any $g \in C[t]$ s.t. $g(A) \equiv 0$.

The minimal polynomial of A, $m_A(t)$, is the unique monic (首一) polynomial of minimum positive degree that annihilates A.

Theorem: $m_A$ is well defined.

(Proof: polynomial division)

Corollary: The minimal polynomial divides the characteristic polynomial and has all the same roots.

Property: TFAE:

1. A diagonalizable
2. m_A splits into distinct linear factors
3. every root of m_A has multiplicity 1
4. any t_0 s.t. m_A(t_0) =0, m^1_A(t_0) != 0

Let monic polynomial $f(t) = t^n + a_{n-1} t^{n-1} + \dots + a_0$, the companion matrix of $f$ is $\begin{bmatrix} 0 & - a_0 \\ I & - a_{n-1} \end{bmatrix}$.

Property: Every monic polynomial is the minimal and characteristic polynomial of its companion.

The set of polynomials in one variable $t$ over $\mathbb{C}$ is a vector space.