15-00-MatricesOfSpecialForm.tex

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\pmtitle{matrices of special form}
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\begin{document}
The aim of the present entry is to list matrices with special properties.

\subsubsection*{Restriction on form}
\begin{itemize}
\item diagonal matrices
\item anti-diagonal matrices
\item triangular matrices
\item block matrices of either form above or of some other form
\item nilpotent matrices
\item elementary matrices
\item Hadamard matrices
\item partly decomposable matrices
\item fully indecomposable matrices
\item nearly decomposable matrices
\item doubly stochastic matrices
\item stochastic matrices
\end{itemize}

In numerical applications, sparse matrices (matrices with few nonzero entries) are important; these can be of any of the forms above.

\subsubsection*{Other restrictions}
\begin{itemize}
\item singular matrices
\item $\operatorname{GL}(n)$: invertible matrices
\item $\operatorname{SU}(n)$: unitary matrices with determinant $1$
\item $\operatorname{O}(n)$: orthogonal matrices
\item $\operatorname{SO}(n)$: 
\PMlinkname{orthogonal matrices with determinant $1$}{RotationMatrix}
\item normal matrices
\item positive definite matrices
\item symmetric matrices, antisymmetric matrices, Hermitian matrices, anti-Hermitian matrices
\item $\operatorname{Sp}(2n)$: symplectic matrices (also called $\operatorname{Sp}(n)$)
\end{itemize}


\subsubsection*{Special matrices}
\begin{itemize}
\item zero matrix
\item identity matrix
\item Hilbert matrix
\item Vandermonde matrix
\item Toeplitz matrix
\item Pascal matrix
\item Cauchy matrix
\item M-matrix
\item magic square
\item Latin square
\end{itemize}
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\end{document}