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Version 7 |
| \PMlinkescapeword{properties} |
\PMlinkescapeword{properties} |
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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.\\ |
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.\\\\ |
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:} |
\textbf{properties:} |
| \begin{itemize} |
\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 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. |
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| \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 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. |
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\item Let $A$ be a square real matrix\textit{(possibly complex)} or 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 $A$.
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\item Let $A$ be a square real matrix\textit{(possibly complex)} or 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 $A$.
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| \item A complex square matrix is diagonal if and only if it is normal, triangular.(see theorem for normal triangular matrices). |
\item A complex square matrix is diagonal if and only if it is normal, triangular.(see theorem for normal triangular matrices). |
| \end{itemize} |
\end{itemize} |
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| \textbf{examples:} |
\textbf{examples:} |
| \begin{itemize} |
\begin{itemize} |
| \item $\begin{pmatrix} |
\item $\begin{pmatrix} |
| a & b \\ |
a & b \\ |
| -b & a \\\end{pmatrix}$ where $a,b \in \mathbb{R}$ |
-b & a \\\end{pmatrix}$ where $a,b \in \mathbb{R}$ |
| \item $\begin{pmatrix} |
\item $\begin{pmatrix} |
| 1 & i \\ |
1 & i \\ |
| -i & 1 \\\end{pmatrix}$ |
-i & 1 \\\end{pmatrix}$ |
| \end{itemize} |
\end{itemize} |
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| \textbf{see also:} |
\textbf{see also:} |
| \begin{itemize} |
\begin{itemize} |
| \item Wikipedia, \PMlinkexternal{normal matrix}{http://www.wikipedia.org/wiki/Normal_matrix} |
\item Wikipedia, \PMlinkexternal{normal matrix}{http://www.wikipedia.org/wiki/Normal_matrix} |
| \end{itemize} |
\end{itemize} |
