Normal Matrix

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 $V$ consisting of eigenvectors of $A$. Then by Theorem 2, $D = [L_A]_{\beta} = Q^{-1}AQ$ where $Q = [I]_{\beta}^{\gamma}$ for the standard ordered basis $\gamma$ for $\mathbb{C}^n$. Note that $Q$ is the matrix whose columns are the vectors in $\beta$. Since $\beta$ is orthonormal, by remark, $Q^*Q = I = QQ^*$. Thus $Q$ is unitary, so $A$ is unitarily equivalent to $D$.
$(\Longleftarrow)$
Suppose that $D = P^*AP$ for some unitary matrix $P$. Note that $A = PDP^*$. Then $AA^* = (PDP^*)(PD^*P^*) = PDD^*P^* = (PD^*P^*)(PDP^*) = A^*A$. $\blacksquare$