Fast Parallel Algorithms for Matrix Reduction to Normal Forms

Gilles Villard
1997 Applicable Algebra in Engineering, Communication and Computing  
We investigate fast parallel algorithms to compute normal forms of matrices and the corresponding transformations. Given a matrix B in M LL (K), where K is an arbitrary commutative field, we establish that computing a similarity transformation P such that F"P\BP is in Frobenius normal form can be done in NC ) . Using a reduction to this first problem, a similar fact is then proved . We get that over concrete fields such as the rationals, these problems are in NC. Using our previous results we
more » ... ve thus established that the problems of computing transformations over a field extension for the Jordan normal form, and transformations over the input field for the Frobenius and the Smith normal form are all in NC ) . As a corollary we establish a polynomial-time sequential algorithm to compute transformations for the Smith form over K [x].
doi:10.1007/s002000050089 fatcat:nq52uowzobh3dasfbkwfrvs76y