Discrete Dynamical Systems행렬 $A \in M_{k \times k}(\mathbb{R})$에 대해서 LdE $X_n = A X_{n-1}, n \in \mathbb{N}$을 discrete dynamical system이라고 부른다. $A$가 대각화 가능하다면 $k$개의 선형독립인 고유벡터 $v_1, ..., v_k$과 각각에 대응되는 고유값 $\lambda_1, ..., \lambda_k$가 존재한다. 이때 해는 $$X_n = c_1 \lambda_1^nv_1 + \cdots + c_k \lambda_k^n v_k$$로 주어진다. 한편 $n$이 점점 커질수록 주어진 해 $X_n$은 각 고유값 $\lambda$의 크기에 따라 그 거동이 달라진다. 만약 모든 고유값의 크..

Linear Difference EquationsDefinition 1. Define a sequence $\{ X_n \in \mathbb{R}^k \mid n \ge 0 \}$ by a matrix equation $$X_n = A X_{n-1}, \quad n \ge 1, \quad \text{with an arbitrary } k \times k \text{ square matrix } A.$$ Such an equation $X_n = A X_{n-1}, n \ge 1$ is called a linear difference equation (LdE) of order $k$.A solution to an LdE $X_n = A X_{n-1}$ is a sequence $\{ X_n \in \mat..

Linear Recurrence RelationsDefinition 1. A sequence $\{x_n\}_{n = 0}^{\infty}$ is said to satisfy a linear recurrence relation (LRR) of order $k$ if $\exists\,a_1,\dots,a_k$ with $a_k\neq 0$ such that $$ x_n=a_1x_{n-1}+\cdots+a_kx_{n-k}, \forall n\ge k. $$번역하면 '선형 점화식'이며, 줄여서 LRR이라고 부른다. 고등학교 교육과정에서는 '수열의 귀납적 정의'정도로 부르는 것 같다. 보통 수열은 일반항으로 주어져서 임의의 $n$번째 항을 공식을 통해 구할 수 있다. 그러나 특정한 규칙으로 정의되는 수열의 경..

Minimal PolynomialDefinition 1. Let $A \in M_{n \times n}(\mathbb{C})$. A minimal polynomial of $A$ is the monic polynomial $m(t)$ of smallest degree such that $m(A) = O$.Remark 1Remark 1. Let $A \in M_{n \times n}(\mathbb{C})$.(i) If $f(t)$ is a polynomial such that $f(A) = O$, then $m(t)$ divides $f(t)$.(ii) There is the unique minimal polynomial of $A$.(iii) $m(t)$ divides the characteristic ..

Exponential MatrixDefinition 1. For $A \in M_{n \times n}(\mathbb{C})$, the exponential matrix of $A$ is defined by $$e^A = \sum_{k=0}^{\infty} \frac{A^k}{k!} = I + \frac{A}{1!} + \frac{A^2}{2!} + \frac{A^3}{3!} + \cdots$$Theorem 1. For $A \in M_{n \times n}(\mathbb{C})$, $e^A$ is well-defined.Proof. Put $M = \max_{i, j} \{ |a_{ij}| \}$ where $a_{ij} = [A_{ij}]$. Note that $$[A^2]_{ij} \le \begi..

Jordan BlockDefinition 1. A matrix of the type of $$J = \begin{bmatrix} \lambda & 1 & & \mathbf{0} \\ & \ddots & \ddots & \\ & & \ddots & 1 \\ \mathbf{0} & & & \lambda \end{bmatrix}$$ is called a Jordan block belonging to $\lambda$.Jordan Canonical FormTheorem 1. Every square matrix $A$ is similar to the Jordan canonical form $J$ of $A$: $$J = \begin{bmatrix} J_1 & & & \mathbf{0} \\ & J_2 & & \..

Positive Definite, SemidefiniteDefinition 1. Let $T \in \mathcal{L}(V)$ where $V$ is a finite-dimensional inner product space. (i) $T$ is said to be positive definite if $T$ is Hermitian and $\langle T(x), x \rangle > 0 \forall x \neq \mathbf{0}$.(ii) $T$ is said to be positive semidefinite if $T$ is Hermitian and $\langle T(x), x \rangle \geq 0, \forall x \neq \mathbf{0}$.Definition 2. Let $A \..

Spectral TheoremTheorem 1. Let $T \in \mathcal{L}(V)$ where $V$ is a finite-dimensional inner product space over $F$ with the distinct eigenvalues $\lambda_1, \cdots, \lambda_k$. Assume that $T$ is normal if $F = \mathbb{C}$ and that $T$ is hermitian if $F = \mathbb{R}$. For each $i (1 \leq i \leq k)$, let $W_i$ be the eigenspace of $T$ corresponding to the eigenvalue $\lambda_i$, and that $T_i$..

Schur's TheoremTheorem 1. Let $T \in \mathcal{L}(V)$ where $V$ is a finite-dimensional inner product space. Then there exists an orthonormal basis $\beta$ for $V$ such that $[T]_{\beta}$ is upper triangular. Proof. Let $n = \dim(V)$. The proof is by the mathematical induction on $n$. If $n = 1$, the result is immediate. So suppose that the theorem is true for $n-1$ where $n-1 \geq 1$. Let $W$ be..

Unitarily, Orthogonally SimilarDefinition 1. (i) Let $A, B \in M_{n \times n}(\mathbb{C})$. Then $A$ and $B$ are unitarily similar if there exists a unitary matrix $U$ such that $A = U^*BU$.(ii) Let $A, B \in M_{n \times n}(\mathbb{R})$. Then $A$ and $B$ are orthogonally similar if there exists an orthogonal matrix $U$ such that $A = U^TBU$.Unitarily, Orthogonally DiagonalizableDefinition 2. (i)..

Unitary, Orthogonal MatrixDefinition 1. Let $A \in M_{n \times n}(F)$. Then $A$ is called a unitary matrix if $A^*A = AA^* = I$ and is called an orthogonal matrix if $A^tA = AA^t = I$.Theorem 1Theorem 1. Let $A \in M_{n \times n}(F)$. TFAE.(i) The column vectors of $A$ are orthonormal.(ii) $A^*A = AA^* = I$.(iii) $\|Ax\| = \|x\|, \forall x \in F^n$.(iv) The row vectors of $A$ are orthonormal.

Unitary, OrthogonalDefinition 1. Let $T \in \mathcal{L}(V)$ where $V$ is a finite-dimensional inner product space over $F$. If $||T(x)|| = ||x||, \forall x \in V$, we call $T$ a unitary operator if $F = \mathbb{C}$ and call $T$ an orthogonal operator if $F = \mathbb{R}$.RemarkRemark. Let $V$ be a finite-dimensional inner product space over $F$ and let $T \in \mathcal{L}(V)$.(i) If $T$ is unitary..

Bessel's InequalityTheorem 1. Let ($V, \langle \cdot, \cdot \rangle$) be an inner product space, and let $S = \{v_1, ..., v_n\}$ be an orthonormal subset of $V$. Then $\forall x \in V$, $$||x||^2 \geq \sum_{i=1}^n |\langle x, v_i \rangle|^2.$$Proof. Let $\langle S \rangle = W$. Then $! \exists y \in W, z \in W^{\perp}$ such that $x = y + z$ by Theorem 1. Thus we have $$||x||^2 = ||y||^2 + ||z||^..

Hermitian MatrixDefintion 1. Let $A \in M_{n \times n}(\mathbb{C})$. (i) $A$ is said to be Hermitian (or self-adjoint) if $A = A^*$.(ii) $A$ is said to be skew-Hermitian if $A^* = -A$.Remark. Let $A \in M_{n \times n}(\mathbb{C})$. (i) For $x, y \in \mathbb{C}^n$, $(Ax) \cdot y = x \cdot (A^*y)$.(ii) If $A$ is real symmetric, then $A$ is Hermitian.(iii) If $A$ is real skew-symmetric, then $A$ is..

Normal MatrixDefintion 1. Let $A \in M_{n \times n}(\mathbb{C})$. We say that $A$ is normal if $AA^* = A^*A$.Remark. Every Hermitian, skew-Hermitian, and unitary matrix is normal.Theorem 1Theorem 1. Let $A \in M_{n \times n}(\mathbb{C})$. Then $A$ is normal $\iff$ $A$ is unitarily diagonalizable.Proof. ($\Longrightarrow$)By Theorem 2, there exists an orthonormal basis for $V$ consisting of eigen..

HermitianDefintion 1. (i) Let $T \in \mathcal{L}(V)$ where $V$ is an inner product space. $T$ is said to be hermitian (or self-adjoint) if $T = T^*$.(ii) Let $A \in M_{n \times n}(\mathbb{C})$. $A$ is said to be hermitian (or self-adjoint) if $A = A^*$.Remark. Let $T \in \mathcal{L}(V)$ where $V$ is an inner product space. (i) $T$ is hermitian $\iff$ $[T]_{\beta}$ is hermitian for some orthonorm..

Normal OperatorDefintion 1. (i) Let $T \in \mathcal{L}(V)$ where $V$ is an inner product space. $T$ is said to be normal if $TT^* = T^*T$.(ii) Let $A \in M_{n \times n}(\mathbb{C})$. $A$ is said to be normal if $AA^* = A^*A$.Remark. Let $T \in \mathcal{L}(V)$ where $V$ is an inner product space. Then $T$ is normal $\iff$ $[T]_{\beta}$ is normal for some orthonormal basis $\beta$ for $V$.Theorem ..

SumDefinition 1. Let $W_1, ..., W_k \leq V$. We define the sum of these subspaces to be the set $\{v_1 + \cdots + v_k \,|\, v_i \in W_i \text{ for } 1 \leq i \leq k\}$, which we denote by $$\sum_{i=1}^k W_i.$$Direct SumDefinition 2. Let $W_1, ..., W_k \leq V$. We call $V$ the direct sum of $W_1, ..., W_k$ and write $$V = \bigoplus_{i=1}^k W_i,$$ if $V = \sum_{i=1}^k W_i$ and $W_j \cap \sum_{i \n..

The Cayley-Hamilton Theorem Theorem 1. (The Cayley-Hamilton Theorem) Let $T \in \mathcal{L}(V)$, and let $f(t)$ be the characteristic polynomial of $T$. (V is finite-dimensional) Then $f(T) = T_0$, the zero transformation.Proof. We need to show that $f(T)(v) = \mathbf{0}, \forall v \in V$. If $v = \mathbf{0}$, it is clear. Suppose that $v \neq \mathbf{0}$. Let $W$ be the $T$-cyclic subspace of $..

The Cyclic SubspaceDefinition 1. Let $T \in \mathcal{L}(V)$, and let a nonzero vector $x \in V$. The subspace $W = \langle x, T(x), T^2(x), ... \rangle$ is called the $T$-cyclic subspace of $V$ generated by $x$.Theorem 1Theorem 1. Let $T \in \mathcal{L}(V)$, and let $W$ be the $T$-cyclic subspace of $V$ generated by $\mathbf{0} \neq x \in V$. Then(a) $W$ is $T$-invariant.(b) Any $T$-invariant su..

The Invariant SubspaceDefinition 1. Let $T \in \mathcal{L}(V)$. Then $W \leq V$ is called a $T$-invariant subspace of $V$ if $T(W) \subseteq W$. $W$의 image가 다시 $W$에 포함될 때 $W$를 $T$-불변 부분공간이라고 부른다. 자명하게 $\{\mathbf{0}\}, V, R(T), N(T), E_{\lambda}$는 $T$-불변 부분공간임을 알 수 있다.The restriction of a Linear OperatorDefinition 2. Let $T \in \mathcal{L}(V)$, and let $W$ be a $T$-invariant subspace of $V$. T..

어떤 선형 연산자 $T$가 주어졌을 때 대각화가능한지 결정하고, 가능하다면 대각화하도록 고유벡터들로 이루어진 기저 $\beta$를 찾는 것이 우리의 목표이다. $T$의 고유값은 특성 다항식 $f(t) = \det (T - tI)$를 풀어서 구할 수 있다. 만약 이를 통해 서로 다른 고유값 $\lambda_1, ..., \lambda_k$를 구했을 때, 이 고유값들에 대응되는 고유벡터들은 $v \in E_{\lambda}$을 이용해서 구할 수 있다. 이제 이 고유벡터들로 기저를 구성해야 하고, 그 방법을 아래의 정리들이 제시해준다. Theorem 1Theorem 1. Let $T \in \mathcal{L}(V)$, and let $\lambda_1, ..., \lambda_k$ be distinct ei..

The MultiplicityDefintion 1. Let $T \in \mathcal{L}(V)$, and let $\lambda$ be an eigenvalue of $T$ with characteristic polynomial $f(t)$. Then (i) The algebric multiplicity of $\lambda$ is the largest positive integer $k$ for which $(t - \lambda)^k$ is a factor of $f(t)$.(ii) The geometric multiplicity of $\lambda$ is $\dim(E_{\lambda})$ where $E_{\lambda}$ is the eigenspace of T belonging to $\..

The EigenspaceDefinition 1. (i) Let $T \in \mathcal{L}(V)$, and let $\lambda$ be an eigenvalue of $T$. The eigenspace of $T$ belonging to $\lambda$ is the set $E_{\lambda} = N(T - \lambda I_V) = \{x \in V \,|\, T(x) = \lambda x\}$. (ii) Let $A \in M_{n \times n}(F)$, and let $\lambda$ be an eigenvalue of $A$. The eigenspace of $A$ belonging to $\lambda$ is the set $E_{\lambda} = N(A - \lambda I_..

이 포스트에서 $V$는 $n$차원 $F$-벡터공간으로 취급한다.Theorem 1Theorem 1. (i) Let $T \in \mathcal{L}(V)$. Then a scalar $\lambda$ is an eigenvalue of $T$ $\Longleftrightarrow$ $\det(T - \lambda I_V) = 0$.(ii) Let $A \in M_{n \times n}(F)$. Then a scalar $\lambda$ is an eigenvalue of $A$ $\Longleftrightarrow$ $\det(A - \lambda I_n) = 0$.Proof. (i) Since $\lambda$ is an eigenvector of $T$, there is an nonzero vector..

이 포스트에서 $V$는 유한차원 $F$-벡터공간으로 취급한다.Definition 1Definition 1. Let $T \in \mathcal{L}(V)$ and let $\beta$ be an ordered basis for $V$.(i) The determinant of $T$ is defined by $\det(T) = \det([T]_{\beta})$.(ii) The trace of $T$ is defined by $\text{tr}(T) = \text{tr}([T]_{\beta})$.Remark. Let $\beta, \gamma$ be ordered bases for $V$. Then We have $[T]_{\gamma} = Q^{-1}[T]_{\beta}Q,$ where $Q = [I_V]..

이 포스트에서 $V$는 유한차원 $F$-벡터공간으로 취급한다.DiagonalizableDefinition 1. (i) Let $T \in \mathcal{L}(V)$. $T$ is said to be diagonalizable if there exists an ordered basis $\beta$ for $V$ such that $[T]_{\beta}$ is a diagonal matrix. (ii) Let $A \in M_{n \times n}(F)$. $A$ is said to be diagonalizable if there exists an ordered basis $\beta$ for $F^n$ such that $\left[ \mathsf{L}_A \right]_{\beta}$ is a dia..

Least Square Solution$A \in M_{m \times n}(F), x \in F^n, b \in F^m$에 대해서 선형연립방정식 $Ax = b$을 생각해보자. 이때 다음의 두 가지 경우의 수가 존재한다.$b \in \mathcal{C}(A)$$b \notin \mathcal{C}(A)$1번의 경우 해 $x \in F^n$이 반드시 존재하므로 그냥 구하면 된다. 그런데 2번의 경우 해가 존재하지 않는다. 이런 경우에라도 울며 겨자먹기로 진짜 해는 아니지만 가장 해에 근접하다고 말할 수 있을만한, 다시 말해 $\|Ax_0 - b\|$를 최소화하는 그러한 $x_0 \in F^n$을 항상 찾을 수 있을까? 이는 대단히 실용적인 문제이며, 주어진 데이터들을 완벽하게 설명할 순 없지만 가장 근사적..

Adjoint of MatrixDefinition 1. Let $A \in M_{m \times n}(F)$. We define the adjoint or conjugate transpose of $A$ to be the $n \times m$ matrix $A^*$ such that $(A^*)_{ij} = \overline{A_{ji}}$ for all $i, j$.Theorem 1Theorem 1. Let $A \in M_{m \times n}(F)$. Then $L_{A^*} = (L_A)^*$.Proof. Note that $[L_{A^*}]_{\gamma}^{\beta} = A^* = ([L_A]_{\beta}^{\gamma})^* = [(L_A)^*]_{\gamma}^{\beta},$ whe..

행렬의 adjoint는 원 행렬의 켤레 전치로 정의되었다. 유사하게 선형 변환의 adjoint를 정의하려고 한다. 어떤 선형 변환 $T$에 대해 $([T]_{\beta}^{\gamma})^* = [U]_{\gamma}^{\beta}$를 만족하는 선형 변환 $U$를 찾고, 그 $U$를 $T$의 adjoint라고 정의하는 것이 자연스러울 것이다.Theorem 1Theorem 1. Let $\mathsf{g} \in V^*$ where $V$ is a finite-dimensional inner product space over $F$. Then $\exists ! \, y \in V$ such that $\mathsf{g}(x) = \langle x, y\rangle, \forall x \in V$.Proo..