Properties of the logarithmic matrix function

The following properties of the logarithmic matrix function are proved in Exercise 3.6  

Equation (3.4.12) holds for all matrices and is the analogue of (3.1.4) for matrix exponentials. Relationships (3.4.13) and (3.4.14) are similar to those of the scalar logarithmic function. Equation (3.4.13) holds for all values of n (not necessarily integers) when matrix powers are defined as 

In Exercise 3.6, the proof given for (3.4.14) is valid only for matrices belonging to the domain where the logarithmic and exponential matrix functions are inverse functions. Finally, (3.4.15) - which follows from (3.4.12) and (3.4.13) - shows that the logarithmic function maps unitary matrices onto anti-Hermitian matrices as expected from the fact that flie exponential function maps anti-Hermitian matrices onto unitary matrices. 

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