Matrix sector functions and their applications to systems theory

The paper presents a new matrix function, the matrix sector function of a square complex matrix A, and its applications to systems theory. Firstly, based on an irrational function of a complex variable λ, a scalar sector function of λ,(λ/n√λn), is defined. Next, a fast algorithm is developed with the help of a circulant matrix for computing the scalar sector function of λ. Then, the scalar sector function of λ is extended to a matrix sector function of A, A(n√An)−1, and to associated partitioned matrix sector functions of A. Finally, applications of these matrix sector functions to the separation of matrix eigenvalues, the determination of A-invariant space, the block diagonalisation of a matrix, and the generalised block partial fraction expansion of a rational matrix are given. It is shown that the well-known matrix sign function of A is a special class of the newly developed matrix sector function of A. It is also shown that the Newton-Raphson type algorithm cannot, in general, be applied to determine the matrix sector function of A.