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..
