Linear-2 3

Diagonalizable matrix and eigenvectors #lemma

\begin{matrix} Let A \in F^{n \times n} \\ A is diagonalizable ⟺ \exists B basis of F^{n} : \forall i \in [1, n] : A v_{i} = λ_{i} v_{i} \\ Proof: \\ ⟸ Let \exists B basis of F^{n} : \forall i \in [1, n] : A v_{i} = λ_{i} v_{i} \\ Let B = {v_{1}, \dots, v_{n}} \\ Let {λ_{i}}_{i \in [1, n]} : \forall i \in [1, n] : A v_{i} = λ_{i} v_{i} \\ Let P = (\begin{matrix} \overset{|}{\underset{|}{v_{1}}} & \dots & \overset{|}{\underset{|}{v_{n}}} \end{matrix}) \in F^{n \times n} \\ B is a linear independence ⟹ r a n k (P) = n ⟹ P is invertible \\ P^{- 1} A P = P^{- 1} (\begin{matrix} \overset{|}{\underset{|}{A v_{1}}} & \dots & \overset{|}{\underset{|}{A v_{n}}} \end{matrix}) = P^{- 1} (\begin{matrix} \overset{|}{\underset{|}{λ_{1} v_{1}}} & \dots & \overset{|}{\underset{|}{λ_{n} v_{n}}} \end{matrix}) = \\ = (\begin{matrix} \overset{|}{\underset{|}{λ_{1} P^{- 1} v_{1}}} & \dots & \overset{|}{\underset{|}{λ_{n} P^{- 1} v_{n}}} \end{matrix}) \\ λ_{i} P^{- 1} v_{i} = λ_{i} P^{- 1} C_{i} (P) = λ_{i} P^{- 1} P e_{i} = λ_{i} e_{i} \\ ⟹ P^{- 1} A P = (\begin{matrix} λ_{1} & 0 & \dots & 0 \\ 0 & λ_{2} & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & 0 \\ 0 & \dots & 0 & λ_{n} \end{matrix}) = D \\ ⟹ Let A be diagonalizable \\ Let D = (\begin{matrix} α_{1} & 0 & \dots & 0 \\ 0 & α_{2} & ⋱ & ⋮ \\ ⋮ & ⋱ & ⋱ & 0 \\ 0 & \dots & 0 & α_{n} \end{matrix}) \\ Let P = (\begin{matrix} \overset{|}{\underset{|}{p_{1}}} & \dots & \overset{|}{\underset{|}{p_{n}}} \end{matrix}) \in F^{n \times n} be invertible \\ D = P^{- 1} A P = P^{- 1} (\begin{matrix} \overset{|}{\underset{|}{A p_{1}}} & \dots & \overset{|}{\underset{|}{A p_{n}}} \end{matrix}) = (\begin{matrix} \overset{|}{\underset{|}{P^{- 1} A p_{1}}} & \dots & \overset{|}{\underset{|}{P^{- 1} A p_{n}}} \end{matrix}) \\ ⟹ \forall i \in [1, n] : C_{i} (D) = P^{- 1} A p_{i} ⟹ α_{i} e_{i} = P^{- 1} A p_{i} \\ ⟹ α_{i} P e_{i} = A p_{i} ⟹ α_{i} p_{i} = A p_{i} \\ ⟹ \forall i \in [1, n] : A p_{i} = α_{i} p_{i} \\ r a n k (P) = n ⟹ {p_{1}, \dots, p_{n}} is a linear independence of size n \\ ⟹ {p_{1}, \dots, p_{n}} is a basis of F^{n} \end{matrix}

Eigenvalues and eigenvectors of a linear transformation #definition

\begin{matrix} Let V be a finitely generated vector space over F \\ Let T : V \to V be a linear transformation \\ λ \in F is called an Eigenvalue of T if \\ \exists v \neq 0 \in F^{n} : T (v) = λ v \\ v is then called an Eigenvector of T in respect to Eigenvalue λ \end{matrix}

Eigenvalues of linear transformation and representation matrix #lemma

\begin{matrix} Let V be a finitely generated vector space over F \\ Let B be a basis of V \\ Let T : V \to V be a linear transformation \\ Then λ is an eigenvalue of T ⟺ λ is an eigenvalue of [T]_{B}^{B} \\ Proof: \\ λ is an eigenvalue of T ⟺ \exists v \neq 0 \in V : T (v) = λ v \\ ⟺ \exists v \neq 0 \in V : [T]_{B}^{B} [v]_{B} = [λ v]_{B} = λ [v]_{B} ⟺ λ is an eigenvalue of [T]_{B}^{B} \end{matrix}

Corollary

\begin{matrix} v is an eigenvector of T ⟺ [v]_{B} is an eigenvector of [T]_{B}^{B} \end{matrix}

Diagonalizable linear operator #definition

\begin{matrix} Let V be a finitely generated vector space over F \\ Let T : V \to V be a linear operator \\ T is then called diagonalizable iff \exists B basis of V : [T]_{B}^{B} is diagonalizable \end{matrix}

Diagonalizable linear operator and representation matrix #lemma

\begin{matrix} Let V be a finitely generated vector space over F \\ Let T : V \to V be a linear operator \\ Let C be a basis of V \\ Then T is diagonalizable ⟺ [T]_{C}^{C} is diagonalizable \\ Proof: \\ ⟹ Let T be diagonalizable \\ ⟹ \exists B : [T]_{B}^{B} is diagonal \\ ⟹ ([I]_{C}^{B})^{- 1} [T]_{C}^{C} [I]_{C}^{B} = [T]_{B}^{B} \\ ⟹ [T]_{C}^{C} is diagonalizable \\ ⟸ Let [T]_{C}^{C} be diagonalizable \\ Let P \in F^{n \times n} be invertible \\ ⟹ \exists B : P = [I]_{C}^{B} \\ ⟹ P^{- 1} [T]_{C}^{C} P = [I]_{B}^{C} [T]_{C}^{C} [I]_{C}^{B} = [T]_{B}^{B} = D \\ ⟹ T is diagonalizable \end{matrix}

Properties of characteristic polynomial #lemma

\begin{matrix} 1. Eigenvalues of A are roots of its characteristic polynomial \\ 2. Characteristic polynomial is a monic polynomial of degree n \\ That is, its leading coefficient is 1 \\ 3. P_{A} (λ) = λ^{n} + a_{n - 1} λ^{n - 1} + \dots + a_{0} \cdot 1 ⟹ {\begin{matrix} a_{n - 1} = - t r (A) \\ a_{0} = (- 1)^{n} | A | \end{matrix} \\ Proof for 2. \\ P_{A} (λ) = | λ I - A | = \prod_{i = 1}^{n} (λ - a_{i i}) + \underset{of degree \leq n - 2}{\underset{⏟}{p (λ)}} = \\ = λ^{n} - a_{11} λ^{n - 1} - \dots - a_{n n} λ^{n - 1} + \underset{of degree \leq n - 2}{\underset{⏟}{p_{1} (λ)}} = \\ = λ^{n} + (- t r (A)) λ^{n - 1} + p_{1} (λ) \\ Proof for 3. \\ a_{0} = P_{A} (0) = | 0 I - A | = | - A | = (- 1)^{n} | A | \end{matrix}

Characteristic polynomial of linear operator #definition

\begin{matrix} Let V be a finitely generated vector space over F \\ Let B be a basis of V \\ Let T : V \to V be a linear operator \\ P_{T} (λ) is called a characteristic polynomial of T \\ And is equal to P_{T} (λ) = P_{[T]_{B}^{B}} (λ) \\ Note: \\ Choice of basis does not mattter \\ \forall B, C : [T]_{B}^{B} \sim [T]_{C}^{C} ⟹ P_{[T]_{B}^{B}} (λ) = P_{[T]_{C}^{C}} (λ) \end{matrix}

Algebraic multiplicity #definition

\begin{matrix} Let A \in F^{n \times n} \\ Let λ_{i} be an eigenvalue of A \\ Maximal degree k such that (λ - λ_{i})^{k} ∣ P_{A} (λ) is then called \\ an algebraic multiplicity of λ_{i} and denoted g_{A} = μ_{A} (λ_{i}) = k \end{matrix}

Geometric multiplicity #definition

\begin{matrix} Let A \in F^{n \times n} \\ Let λ_{i} be an eigenvalue of A \\ \dim (N (λ_{i} I - A)) is then called a geometric multiplicity of λ_{i} \\ And is denoted k_{λ} = γ_{A} (λ_{i}) = \dim (N (λ_{i} I - A)) \end{matrix}