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..
