I was using the sage's eigenvalues(), but the function takes so much time (exponential growth) for a higher n*n dimension. I would appreciate it if anyone knows is there any shortcut for hollow matrices, so I can make an efficient algorithm. Thanks in Advance!
Generic Structure of a Hollow Matrix: \begin{pmatrix}0&\ast &&\ast &\ast \\\ast &0&&\ast &\ast \\&&\ddots \\\ast &\ast &&0&\ast \\\ast &\ast &&\ast &0\end{pmatrix} Example of a Hollow matrix: \begin{pmatrix}0&2&6&{\frac {1}{3}}&4\\2&0&56&8&0\\9&4&0&2&93\\1&2&4&0&4\\3&9&83&8&0\end{pmatrix} The difference between the example to mine is that the size is quite extensive; I intend to come up with an efficient algorithm to find eigenvalues() quickly, reducing time complexity. I would appreciate it if any mathematician could cite any research papers, theorems, or anything that might be helpful to my goal.
The hollowness has nothing to do with anything. Sage has a bad algorithm, but in general, determinant of an integer matrix can be computed in time only slightly worse than for floating point (one way is to use chinese remaindering: compute the determinant mod a bunch of primes, then reconstitute). Yes, I know your matrix is not integer, but clear the denominators first.