The Matrix Unwinding Function, with an Application to Computing the Matrix Exponential

A new matrix function corresponding to the scalar unwinding number of Corless, Hare, and Jeffrey is introduced. This matrix unwinding function, U , is shown to be a valuable tool for deriving identities involving the matrix logarithm and fractional matrix powers, revealing, for example, the precise relation between logAα and α logA. The unwinding function is also shown to be closely connected with the matrix sign function. An algorithm for computing the unwinding function based on the Schur–Parlett method with a special reordering is proposed. It is shown that matrix argument reduction using the function mod(A) = A−2πiU(A), which has eigenvalues with imaginary parts in the interval (−π, π] and for which eA = emod(A), can give significant computational savings in the evaluation of the exponential by scaling and squaring algorithms.