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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 Â¿, a scalar sector function of Â¿,(Â¿/nÂ¿Â¿n), is defined. Next, a fast algorithm is developed with the help of a circulant matrix for computing the scalar sector function of Â¿. Then, the scalar sector function of Â¿ is extended to a matrix sector function of A, A(nÂ¿An)Â¿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.