정규 행렬

선형대수학에서 정규 행렬(正規行列, 영어: normal matrix)은 스스로의 켤레 전치와 가환하는 정사각 행렬이다.^[1]^[2]

정의

복소수 $n\times n$ 정사각 행렬 $N$ 에 대하여, 다음 두 조건이 서로 동치이며, 이를 만족시키는 $N$ 을 정규 행렬이라고 한다.^[1]^:664–665

$NN^{\dagger }=N^{\dagger }N$ $NN^{\dagger }=N^{\dagger }N$ ( $N^{\dagger }$ $N^{\dagger }$ 는 켤레 전치).
- 실수 행렬의 경우 이 조건은 $NN^{\top }=N^{\top }N$ 과 동치이다 ( $N^{\top }$ 은 전치 행렬).
(유니터리 대각화 가능성) $U^{-1}NU$ $U^{-1}NU$ 가 대각 행렬이 되는 유니터리 행렬 $U\in \operatorname {U} (n)$ $U\in \operatorname {U} (n)$ 이 존재한다.
- 특히, 정규 행렬의 고유 공간들은 서로 직교한다.
- 이 조건은 실수 행렬에서 직교 대각화 가능성으로 대체할 수 없다. (예를 들어, 0도 또는 180도가 아닌 회전 행렬은 실수 직교 행렬이며, 특히 정규 행렬이지만, 실수 고윳값을 가지지 않는다.)

성질

연산에 대한 닫힘

정규 행렬 $N$ 및 자연수 $k\in \mathbb {N}$ 및 복소수 $a\in \mathbb {C}$ 에 대하여, $aN$ 과 $N^{k}$ 역시 정규 행렬이다. 가역 정규 행렬 $N$ 에 대하여, $N^{-1}$ 역시 정규 행렬이다. 그러나 두 정규 행렬의 합과 곱은 정규 행렬일 필요가 없다.

충분 조건

다음 복소수 정사각 행렬들은 모두 정규 행렬이다.^[1]^:664–665

복소수 정사각 행렬에 대하여, 다음 세 조건이 서로 동치이다.^[2]^{:315–316, §8.5, Theorem 20}

상삼각 정규 행렬이다.
하삼각 정규 행렬이다.
대각 행렬이다.

유리 표준형

복소수 $n\times n$ 정사각 행렬 $N$ 이 정규 행렬이라면, $U^{-1}NU$ 가 유리 표준형이 되는 유니터리 행렬 $U\in \operatorname {U} (n)$ 이 존재한다.^[2]^{:356, §9.6, Corollary} 만약 추가로 $N$ 의 모든 성분이 실수일 경우, $Q^{-1}NQ$ 가 유리 표준형이 되는 실수 직교 행렬 $Q\in \operatorname {O} (n;\mathbb {R} )$ 가 존재한다.^[2]^{:356, §9.6, Corollary}

같이 보기

정규 작용소

각주

↑ ^가 ^나 ^다 Howard Anton, 이장우 역, 《알기쉬운 선형대수》, 범한서적주식회사, 2006
↑ ^가 ^나 ^다 ^라 Hoffman, Kenneth; Kunze, Ray (1971). 《Linear algebra》 (영어) 2판. Englewood Cliffs, N. J.: Prentice-Hall. ISBN 0-13-536797-2. MR 0276251. Zbl 0212.36601. IA LinearAlgebraHoffmanAndKunze.

외부 링크

“Normal matrix”. 《Encyclopedia of Mathematics》 (영어). Springer-Verlag. 2001. ISBN 978-1-55608-010-4.
Weisstein, Eric Wolfgang. “Normal matrix”. 《Wolfram MathWorld》 (영어). Wolfram Research.

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