##### Sesión Matemática Discreta

Diciembre 15, 16:30 ~ 16:50

## Drazin Inverse of singular $circ(0,a,0,\dots,0,b)$.

### Panelo, Cristian

Let $C$ be a singular symmetric matrix of order $n$. The Drazin inverse of $C$ is a matrix $D$ such that \begin{enumerate} t\item $C\, C^{D}=C^{D}C$, t\item $C\, C^{D} C=C$, t\item $C^{D} C \, C^{D}=C^{D}$. \end{enumerate} see \cite{davis2012circulant}. Let $A=\left( a_{ij}\right)$ be a matrix of order $n$. We associate with $A$ a digraph $D\left( A\right)$ with $n$ vertices, labeled by $\{1,\dots, n\}$. The vertices of $D\left( A\right)$ are denoted by $1,2,\dots,n$. There is an edge from vertex $i$ to vertex $j$ of weight $a_{ij}$ for each $i,j=1,2,\dots,n$. The circulant matrix $circ(0,a,0,\dots,0,b)$ of order $n$ is denoted by $C_{n}(a,b)$. \textbf{Proposition 1: }The circulant matrix $C_n(a,b)$ is singular if and only if either $a=b$ with $n \equiv 0 \ mod \, 4$ or $a=-b$. There are two natural distance in $D(C_n(a,b))$, one clockwise, and one counter-clockwise. Let $j$ be a vertex of $D(C_n(a,b))$, distances from $1$ to $j$ are: \begin{align*} d^{+}(j)&=j-1\\ d^{-}(j)&=n-j+1 \end{align*} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textbf{Theorem 1: } If $n\equiv 0 \mod 4$, then $C_{n}^{D}(a,a)= \dfrac{1}{na} circ\left( \delta_{1},\delta_{2},\dots,\delta_{n}\right)$ where for $1 \leq j \leq n$ $\delta_{j}= \left( \left( -1\right) ^{\left\lfloor d^{+}\left( j\right) /2\right\rfloor }\left\lfloor d^{-}\left( j\right)/2\right\rfloor +\left( -1\right) ^{\left\lfloor d^{-}\left( j\right)/2\right\rfloor }\left\lfloor d^{+}\left( j\right) /2\right\rfloor \right) \!\!\!\! \mod \!\!\left( 2k\right)$ and $C_{n}^{D}(a,-a)= \dfrac{1}{na} circ\left( \delta_{1}^{\prime },\delta_{2}^{\prime },\dots,\delta_{n}^{\prime }\right)$ where for $1 \leq j \leq n$ $\delta_{j}^{\prime }= sgn\left( d^{+}\left( j\right) - d^{-}\left( j\right) \right) \left\vert \delta_{j}\right\vert$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textbf{Theorem 2: } If $n\equiv 1 \mod 2$, then $C_{n}^{D}(a,-a)=\dfrac{1}{na} circ\left( \delta_{1},\delta_{2},\dots,\delta_{n}\right)$ where for $1 \leq j \leq n$ $\delta_{j}=\left\{ \begin{array}{rl} \dfrac{d^{+}\left( j\right) }{2}& \text{if } d^{+}(j) \equiv 0 \mod 2,\\ {} & {} \\ -\dfrac{d^{-}\left( j\right) }{2} & \text{otherwise}. \end{array} \right.$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textbf{Theorem 3: } If $n\equiv 2 \mod 4$, then $C_{n}^{D}(a,a)= \dfrac{2}{na} circ\left( \delta_{1},\delta_{2},\dots,\delta_{n}\right)$ where for $1 \leq j \leq n$ $\delta_{j}= \left\{ \begin{array}{cl} \dfrac{d^{+}\left( j\right) - d^{-}\left( j\right) }{4} & \text{if } d^{+}\left( j\right) \equiv 1 \mod 2,\\ {} & {}\\ 0 & \text{otherwise}. \end{array} \right.$ \begin{thebibliography}{9} t %\bibitem{BrualdiBook}{ %Richard A. Brualdi and Dragos Cvetkovic. %\textit{A Combinatorial Approach to Matrix Theory and Its Applications 2009}.} \bibitem{brualdi2008combinatorial}{\textit{A combinatorial approach to matrix theory and its applications}, Brualdi, Richard A and Cvetkovic, Dragos 2008. CRC press.} \bibitem{davis2012circulant}{\textit{Circulant matrices}, Davis, Philip J. 2012. American Mathematical Soc.} \end{thebibliography}

Autores: Jaume, Daniel A. / Panelo, Cristian.