Normal Operator Defintion 1. Let $T \in \mathcal{L}(V)$ where $V$ is an inner product space. We say that $T$ is normal if $TT^* = T^*T$. 위와 같은 조건을 만족시켰을 때 선형 연산자가 normal, 즉 정규하다고 부른다. 자명하게 선형 연산자 $T$가 normal일 조건은 $[T]_{\beta}$가 normal일 조건과 동치이다. ($\beta$는 orthonormal basis) Theorem 1 Theorem 1. Let $T$ be a normal operator on $V$ where $V$ is an inner product space. Then the following statements..
Schur's Theorem Theorem 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 $W$ b..
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. Adjoint of Linear Transformation Definition 1. Let $T \in \mathcal{L}(V, W)$ where $V$ and $W$ are finite-dimensional inner product space with inner products $\langle \cdot, \cdot \rangle _1$ and $\langle ..
Bessel's Inequality Theorem 1. Let ($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 $$||x||^2 = ||y||^2 + ||z|..
Sum Definition 1. Let $W_1, ..., W_k \leq V$. We define the sum of these subspaces to be the set $\{v_1 + \cdots + v_k \,|\, v_i \in W_i \text{ for } 1 \leq i \leq k\}$, which we denote by $$\sum_{i=1}^k W_i.$$ Direct Sum Definition 2. Let $W_1, ..., W_k \leq V$. We call $V$ the direct sum of $W_1, ..., W_k$ and write $$V = \bigoplus_{i=1}^k W_i,$$ if $V = \sum_{i=1}^k W_i$ and $W_j \cap \sum_{i..
