Definition of the logarithmic matrix function

For a nonsingular matrix A that can be diagonalized as in (3.4.2), we define the logarithmic matrix function log A as 

The evaluation of log A is thus carried out in terms of the complex scalar logarithmic function. For a complex number z, the logarithm is obtained from the polar representation 

The logarithmic matrix function log A is thus the true inverse function of the exponential expA in the neighbourhood of 1, mapping matrices in the neighbourhood of 1 onto matrices in the neighbourhood of 0. 

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