Linear-2 6

Minimal polynomial #definition

\begin{matrix} Let A \in F^{n \times n} \\ Let f (x) \in F [x] \\ f is called a minimal polynomial of A if \\ A is it’s root and there are no such polynomials of smaller degree \\ Minimal matrix polynomial is denoted as m_{A} (x) \\ Note: minimal polynomial is always monic \end{matrix}

Existence and uniqueness of minimal polynomial #lemma

\begin{matrix} Let A \in F^{n \times n} \\ \exists! m_{A} (x) : m_{A} (A) = 0 \\ Proof: \\ By Cayley-Hamilton theorem: P_{A} (A) = 0 \\ P_{A} (x) is of degree n \\ Let f = P_{A} (x) \\ Let us make n choices: \\ Choice 1. ∄ \deg m_{A} < \deg f \\ Choice 2. \exists m_{A} \in F [x] : \deg m_{A} < \deg f ⟹ f = m_{A} \\ After n choices, f definitely contains the minimal polynomial \\ Let f, g be minimal polynomials of A \\ f (x) = x^{k} + \sum_{i = 1}^{k - 1} α_{i} x^{i} \\ g (x) = x^{k} + \sum_{i = 1}^{k - 1} β_{i} x^{i} \\ ⟹ f (x) - g (x) \in F_{t} [x] : t \leq k - 1, \frac{1}{α_{t} - β_{t}} (f - g) (A) = 0 - Contradiction! \\ ⟹ \exists! f minimal polynomial of A \end{matrix}

Minimal polynomial divides any polynomial with matrix as a root #lemma

\begin{matrix} Let A \in F^{n \times n} \\ Let f (x) \in F [x] : f (A) = 0 \\ Then m_{A} (x) ∣ f (x) \\ Proof: \\ Case 0. f = 0 and we are done \\ Case 1. f \neq 0 \\ \deg f \geq \deg m_{A} \\ ⟹ \exists q, r \in F [x] : f (x) = q (x) m_{A} (x) + r (x) \\ \deg r (x) < \deg m_{A} \\ f (A) = q (A) \underset{= 0}{\underset{⏟}{m_{A} (A)}} + r (A) = 0 \\ ⟹ r (A) = 0 ⟹ {\begin{matrix} r = 0 \\ \frac{1}{α} r is a minimal polynomial \end{matrix} \\ ⟹ r = 0 ⟹ m_{A} ∣ f \\ Corollary: \\ m_{A} ∣ P_{A} \\ ⟹ Roots of m_{A} (x) are roots of P_{A} (x) and eigenvalues of A \end{matrix}

Characteristic polynomial divides any polynomial to the power of n with matrix as a root #lemma

\begin{matrix} Let A \in F^{n \times n} \\ Let f (x) \in F [x] : f (A) = 0, \deg f \leq n \\ Then P_{A} ∣ f^{n} \\ Proof: \\ \dots \\ Corollary: \\ P_{A} ∣ m_{A}^{n} \end{matrix}

Corollary of two lemmas above

\begin{matrix} Minimal polynomial contains all irreducible factors of P_{A} \\ at least once and at most algebraic multiplicity of each factor \\ ⟹ All eigenvalues of A are roots of m_{A} \end{matrix}

Jordan block #definition

\begin{matrix} Matrix A is called a Jordan block with element α if \\ A \in F^{n \times n} : A_{i j} = {\begin{matrix} α & i = j \\ 1 & i = j - 1 \\ 0 & otherwise \end{matrix} \\ Jordan block is denoted as J_{n} (α) \\ P_{J_{n} (α)} (λ) = (λ - α)^{n} \\ μ_{J_{n} (α)} (α) = n \\ γ_{J_{n} (α)} (α) = 1 \\ Useful property: \\ (J_{n} (0)^{k})_{i j} = {\begin{matrix} 1 & i = j - k \\ 0 & otherwise \end{matrix} \\ m_{J_{n} (α)} (λ) = (λ - α)^{k}, 1 \leq k \leq n \\ ⟹ m_{J_{n} (α)} (J_{n} (α)) = J_{n} (0)^{k} \\ m_{J_{n} (α)} = 0 ⟹ k = n ⟹ m_{J_{n} (α)} = P_{J_{n} (α)} \end{matrix}

Jordan form #definition

\begin{matrix} Let A \in F^{n \times n} \\ A is said to be a matrix in Jordan form if \\ A can be written as a diagonal block matrix \\ where each block on the diagonal is a Jordan block and all other blocks are 0 \\ e.g. A = (\begin{matrix} J_{2} (3) & 0 & 0 \\ 0 & J_{1} (3) & 0 \\ 0 & 0 & J_{3} (5) \end{matrix}) \in F^{6 \times 6} \\ The common notation is: A = J_{2} (3) \oplus J_{1} (3) \oplus J_{3} (5) \end{matrix}

Jordan decomposition theorem #theorem

\begin{matrix} Let A \in F^{n \times n} \\ Then \begin{matrix} 1. & A \sim A_{J} ⟺ P_{A} is factorizable into linear factors over F \\ 2. & A_{J} is unique up to the order of Jordan blocks \\ 3. & μ_{A} (α) is the sum of sizes of Jordan blocks corresponding to eigenvalue α \\ 4. & γ_{A} (α) is the number of Jordan blocks corresponding to eigenvalue α \\ 5. & Algebraic multiplicity of α in the minimal polynomial \\ is the largest size of Jordan block corresponding to eigenvalue α \end{matrix} \end{matrix}

\begin{matrix} Example: \\ P_{A} (λ) = (λ - 3)^{5} (λ - 1) \\ m_{A} (λ) = (λ - 3)^{2} (λ - 1) \\ γ_{A} (3) = 3 \\ ⟹ A_{J} = (\begin{matrix} J_{1} (1) & 0 & 0 & 0 \\ 0 & J_{1} (3) & 0 & 0 \\ 0 & 0 & J_{2} (3) & 0 \\ 0 & 0 & 0 & J_{2} (3) \end{matrix}) \end{matrix}

Diagonalization and minimal polynomial #theorem

\begin{matrix} Let A \in F^{n \times n} \\ A \sim D ⟺ m_{A} is factorizable into distinct linear factors \\ Proof: \\ ⟹ Let A \sim D \\ ⟹ P_{A} (λ) = \prod_{i = 1}^{k} (λ - α_{i})^{μ_{A} (α_{i})} \\ ⟹ m_{A} (λ) = \prod_{i = 1}^{k} (λ - α_{i})^{t_{i}}, t_{i} \leq μ_{A} (α_{i}) \\ A_{J} = D \\ ⟹ Largest Jordan block corresponding to any eigenvalue of A is of size 1 \\ ⟹ \forall i \in [1, k] : t_{i} = 1 ⟹ m_{A} (λ) = \prod_{i = 1}^{k} (λ - α_{i}) \\ ⟸ Let m_{A} be factorizable into distinct linear factors \\ ⟹ m_{A} (λ) = \prod_{i = 1}^{k} (λ - α_{i}) \\ ⟹ P_{A} (λ) = \prod_{i = 1}^{k} (λ - α_{i})^{μ_{A} (α_{i})} ⟹ A \sim A_{J} \\ ⟹ Largest Jordan block corresponding to any eigenvalue of A is of size 1 \\ ⟹ A_{J} = D ⟹ A \sim D \end{matrix}