Fast tridiagonalization of (p, q)-pentadiagonal matrices and its applications

(p, q)-Pentadiagonal matrices have attracted considerable attention in the past few years, which are one of the generalizations of pentadiagonal matrices. In the current paper, we present an algorithm for tridiagonalization of (p, q)-pentadiagonal matrices using permutation matrices. Moreover, we give the explicit representations of the permutation matrices and describe the nonzero structure of the obtained tridiagonal matrix. As applications, the tridiagonalization provides us with some useful results such as a breakdown-free algorithm of the matrix determinants and permanents, efficient parallel matrix multiplications and integer powers, and a property of the (p, q)-pentadiagonal matrix over the finite field. Some other applications on numerical linear algebra are also discussed.