Abstract:
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.
Keywords:logarithm of matrices, Drazin inverse of matrices, polynomials.
