I would like to find the largest eigenvalue of a sparse matrix by hand- this is part of analyzing a mathematical model for infectious diseases. The nonzero entries are very complicated - hence Maple took 6 minutes and 670MB of memory to find the eigenvalues.
Is there some general way (block matrices?) to find the eigenvalues of the following matrix:
\begin{pmatrix} J_{1,1} & J_{1,2} & J_{1,3} & J_{1,4} & J_{1,5} & J_{1,6} & J_{1,7} & J_{1,8} & J_{1,9} & J_{1,10} & J_{1,11} & J_{1,12} \\ 0 & J_{2,2} & 0 & J_{2,4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ J_{3,1} & J_{3,2} & J_{3,3} & J_{3,4} & J_{3,5} & J_{3,6} & J_{3,7} & J_{3,8} & J_{3,9} & J_{3,10} & J_{3,11} & J_{3,12} \\ 0 & J_{4,2} & 0 & J_{4,4} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & J_{5,2} & 0 & 0 & J_{5,5} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & J_{6,6} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & J_{7,6} & J_{7,7} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & J_{8,7} & J_{8,8} & J_{8,9} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & J_{9,5} & 0 & 0 & 0 & 0 & J_{9,10} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & J_{10,6} & 0 & 0 & J_{10,9} & J_{10,10} & 0 & 0 \\ 0 & J_{11,2} & 0 & J_{11,4} & 0 & 0 & 0 & 0 & 0 & J_{11,10} & J_{11,11} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & J_{12,9} & 0 & J_{12,11} & J_{12,12} \\ \end{pmatrix}
Thanks for any help.