ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

20th Conference of the International Linear Algebra Society (ILAS)

20th ILAS Conference

Contributed talks on Graphs and Networks

Tue 15:00–15:30, Room AV 01.12, Chair: Margarida Mitjana
The Green's function of a weighted $n$–cycle
Margarida Mitjana (Universitat Polit├Ęcnica de Catalunya)
Joint work with Ángeles Carmona (UPC); Andrés M. Encinas (UPC); Silvia Gago (UPC); M. José Jiménez (UPC)

The periodicity of problems in mathematics and applied science leads to the solution of linear systems that involve circulant coefficient matrices. In this work, we analyze a type of circulant matrices namely ${\sf A}={\sf Circ}(a,b,c,\ldots,c,b)$. It turns out that ${\sf A}$ is nothing but the combinatorial Laplacian of the $n$-cycle when $a=2, b=-1$ and $c=0$ or, more generally, for any $q\in \mathbb{R}$, the matrix associated with the Scrödinger operator on the cycle with constant potential $2(q-1)$. Hence, its inverse is the Green's function of the weighted $n$-cycle. The inversion of circulant matrices strongly connects with the resolution of second order difference equations with constant coefficients. Using this approach, we can give a necessary and sufficient condition for the invertibility of matrix ${\sf A}$. It is known that, when exists, the inverse is also a circulant matrix. In this case, we explicitly give a closed formula for the expression of the coefficients of ${\sf A}^{-1}$.

Besides, we give conditions for the invertibility of circulant matrices associated with combinatorial structures such as ${\sf A}={\sf Circ}(a, a+b(n-1), \ldots,a+jb(n-j), \ldots,a+b(n-1))$ or ${\sf A}={\sf Circ}(a,a,b,b,a,a, \ldots, a,a,b,b,a)$.

The case $c=0$ was solved by O. Rojo assuming the condition $|a| > 2|b| > 0$; that is when ${\sf A}$ is a strictly diagonally dominant matrix. In this work we derive the inverse of a general symmetric circulant tridiagonal matrix, without assuming the hypothesis of diagonally dominance.

Acknowledgements: Supported by the Spanish Research Council under project MTM2014-60450-R