This paper presents some new definitions of the real and imaginary parts and the associated amplitude and phase of a real or complex matrix. Computational methods, which utilize the properties of the matrix sign function and the principal nth root of a complex matrix, are given for finding these quantities. A geometric series method is newly developed for finding the approximant of the matrix-valued function of tan−1(X), which is the principal branch of the arc tangent of the matrixX. Several illustrative examples are presented.