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C-Matrix

Any Symmetric Matrix ( ${\hbox{\sf A}}^{\rm T} = {\hbox{\sf A}}$) or Skew Symmetric Matrix ( ${\hbox{\sf A}}^{\rm T}=-{\hbox{\sf A}}$) ${\hbox{\sf C}}_n$ with diagonal elements 0 and others $\pm 1$ satisfying

\begin{displaymath}
{\hbox{\sf C}}{\hbox{\sf C}}^{\rm T} = (n-1){\hbox{\sf I}},
\end{displaymath}

where I is the Identity Matrix, is known as a $C$-matrix (Ball and Coxeter 1987). Examples include
$\displaystyle {\hbox{\sf C}}_4$ $\textstyle =$ $\displaystyle \left[\begin{array}{cccc}0 & + & + & + \\  - & 0 & - & + \\  - & + & 0 & - \\  - & - & + & 0\end{array}\right]$  
$\displaystyle {\hbox{\sf C}}_6$ $\textstyle =$ $\displaystyle \left[\begin{array}{ccccccccccccccccccccccccccccccc}0 & + & + & +...
...& - & + & 0 & + \nonumber\\
+ & + & - & - & + & 0 \end{array}\right].\nonumber$  


References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 308-309, 1987.




© 1996-9 Eric W. Weisstein
1999-05-26