normal matrixNormalMatrix2003-06-13 17:18:112006-08-02 08:55:19Definition% this is the default PlanetMath preamble. as your knowledge
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A complex matrix $A \in \mathbb{C}^{n\times n}$ is said to be \emph{normal} if $A^\ast A = AA^\ast$ where $^\ast$ denotes the conjugate transpose.\\
Similarly for a real matrix $A \in \mathbb{R}^{n\times n}$ is said to be \emph{normal} if $A^TA=AA^T$ where $T$ denotes the transpose.\\\\
\textbf{properties:}
\begin{itemize}
\item Equivalently a complex matrix $A \in \mathbb{C}^{n\times n}$ is said to be \emph{normal} if it satisfies $[A,A^\ast]=0$ where $[,]$ is the commutator bracket.
\item Equivalently a real matrix $A \in \mathbb{R}^{n\times n}$ is said to be \emph{normal} if it satisfies $[A,A^T]=0$ where $[,]$ is the commutator bracket.
\item Let $A$ be a square complex matrix of order $n$. It follows from Schur's inequality that if $A$ is a normal matrix then $\sum_{i=1}^n |\lambda_i|^2 = \operatorname{trace} A^\ast A$ where $^\ast$ is the conjugate transpose and $\lambda_i$ are the eigenvalues of $A$.
\item A complex square matrix is diagonal if and only if it is normal, triangular.(see theorem for normal triangular matrices).
\end{itemize}
\textbf{examples:}
\begin{itemize}
\item $\begin{pmatrix}
a & b \\
-b & a \\\end{pmatrix}$ where $a,b \in \mathbb{R}$
\item $\begin{pmatrix}
1 & i \\
-i & 1 \\\end{pmatrix}$
\end{itemize}
\textbf{see also:}
\begin{itemize}
\item Wikipedia, \PMlinkexternal{normal matrix}{http://www.wikipedia.org/wiki/Normal_matrix}
\end{itemize}