$\mathcal{P}$-Canonical form of the Drazin inverse of matrices, the matrix logarithm, and the principal matrix logarithm

We provide new closed-form expressions for evaluating both the matrix logarithm and its positive integer powers. As a consequence, elegant compact formulas for the principal matrix logarithm and its positive integer powers can be easily obtained. We know that the computation of the matrix logarithm is more complicated and reveals significant difficulties, we reduce these difficulties to a simple problem of determining the standard form of some polynomials. Our results have the advantage of being general and direct. The attractive feature of the proposed approach lies in the possibility of choosing in advance the eigenvalues of logarithms of a matrix and therefore readily obtaining the principal matrix logarithm. Certainly, these interesting results may have a variety of intriguing perspectives in diverse areas of mathematics and natural sciences, particularly in the contexts where the matrix logarithm has proven to be extremely valuable. In addition, significant compact formulas for the arbitrary positive powers of the Drazin inverse are presented.