On O ( n 2 log n ) algorithms for n×n matrix operations

If (without loss of generality) n = p, where p is prime, divide and conquer Fourier transforms using O(nlogn) operations reduce multiplying, or inverting nonsingular, complex n×n matrices to abelian group algebra convolutions. If M is a complex 2×2 matrix, constructing a unitary matrix T and an upper triangular matrix TMT reduces to n(n−1)/2 such constructions in which a 2×2 matrix μ is transformed to an upper triangular matrix τμτ by a unitary matrix τ that represents a quaternion. The diagonal elements of TMT are the eigenvalues of M and, if M is normal, TMT is diagonal and the columns of T are then a complete orthonormal set of eigenvectors. So there is also an algorithm for the classical problem of solving polynomial equations. © 2009 University of Newcastle upon Tyne. Printed and published by the University of Newcastle upon Tyne, Computing Science, Claremont Tower, Claremont Road, Newcastle upon Tyne, NE1 7RU, England. Bibliographical details