Spectral Theorem Theorem 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..

Orthogonal Projection Definition 1. Let $T \in \mathcal{L}(V)$ be a projection, where $V$ is an inner product space. We say that $T$ is an orthogonal projection if $R(T)^{\perp} = N(T)$ and $N(T)^{\perp} = R(T)$. Remark Remark. (a) If $V$ is finite-dimensional, by Theorem 2, we need only assume that one of the preceding conditions holds. (b) If $W \leq V$ is finite-dimensional, then there exists..

Projection Definition 1. Let $W_1, \cdots, W_k \leq V$ such that $V = \bigoplus_{i=1}^k W_i$, and let $T \in \mathcal{L}(V)$ where $V$ is a vector space. Then $T$ is the projection on $W_j$ if, whenever $x = x_1 + \cdots + x_k$ with $x_i \in W_i (i = 1, \cdots, k)$, we have $T(x) = x_j$. Theorem 1 Theorem 1. Let $W_1, \cdots, W_k \leq V$ such that $V = \bigoplus_{i=1}^k W_i$ where $V$ is a vecto..

Hermitian Matrix Defintion 1. Let $A \in M_{n \times n}(F)$. We say that $A$ is hermitian (or self-adjoint) if $A = A^*$. 선형 연산자가 hermitian일 조건과 동일하게 hermitian인 행렬을 정의할 수 있다. 또한 Theorem 1의 행렬 버전을 말할 수 있다. Theorem 1 Theorem 1. Let $A \in M_{n \times n}(\mathbb{R})$. Then $A$ is hermitian $\iff$ $A$ is orthogonally equivalent to a real diagonal matrix. Proof. The proof is similar to the proof of T..

Normal Matrix Defintion 1. Let $A \in M_{n \times n}(F)$. We say that $A$ is normal if $AA^* = A^*A$. 선형 연산자 $T$가 normal일 조건과 동일하게 normal인 행렬을 정의할 수 있다. 또한 Theorem 2의 행렬 버전을 말할 수 있다. Theorem 1 Theorem 1. Let $A \in M_{n \times n}(\mathbb{C})$. Then $A$ is normal $\iff$ $A$ is unitarily equivalent to a diagonal matrix. Proof. ($\Longrightarrow$) By Theorem 2, there exists an orthonormal basis for..

Unitary, Orthogonal Matrix Definition 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 1에 근거해 unitary 혹은 orthogonal 행렬의 정의를 위와 같이 할 수 있다. Remark Remark. $AA^* = I$ [$A^*A = I$] $\iff$ the rows [columns] of $A$ form an orthonormal basis for $F^n$. ($\because$) $\delta_{ij} = I_{Ij} = (AA^*)_..

Unitarily, Orthogonally Equivalent Definition 1. Let $A, B \in M_{n \times n}(\mathbb{C})$ [$M_{n \times n}(\mathbb{R})$]. Then $A$ and $B$ are unitarily equivalent [orthogonally equivalent] if there exists a unitary [orthogonal] matrix $P$ such that $A = P^*BP$ [$A = P^tBP$].

Unitary, Orthogonal Definition 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 $F = \mathbb{R}$. 유한차원의 경우 unitary, 혹은 othogonal, 즉 유니터리 혹은 직교 연산자라고 부르며, 무한차원의 경우 metric을 보존한다는 점을 강조하기 위해 isometry라고 부른다. 자명하게 선형 연산자 $T$가 un..

Metric Definition 1. Given a nonempty set $X$, metric is a function $d: X \times X \rightarrow [0, \infty)$ such that $\forall x, y, z \in X$, the following hold: (a) $d(x, y) = 0 \Longleftrightarrow x = y$ (b) $d(x, y) = d(y, x)$ (c) $d(x, z) \leq d(x, y) + d(y, z)$ Then $(X, d)$ is called a metric space. 위와 같은 성질을 만족하는 함수 $d$를 metric, 즉 거리라고 부른다.

Hermitian Defintion 1. Let $T \in \mathcal{L}(V)$ where $V$ is an inner product space. We say that $T$ is hermitian (or self-adjoint) if $T = T^*$. 위와 같은 조건을 만족시켰을 때 선형 연산자가 hermitian이라고 부른다. 자명하게 선형 연산자 $T$가 hermitian일 조건은 $[T]_{\beta}$가 hermitian일 조건과 동치이다. ($\beta$는 orthonormal basis) 선형 연산자가 normal일 조건을 생각해본다면, hermitian이면 normal임을 쉽게 알 수 있다. Lemma Lemma. Let $T$ be a hermitian operator on a..