Matrix Logarithm VI

The natural logarithm is the inverse operation of the exponential. The following equation defines the natural logarithm of a matrix A: e^B = A, where matrix B is the logarithm of matrix A. A matrix has a logarithm if and only if its inverse matrix exists. For a real matrix A, its logarithm matrix B can be complex, and the conjugate of matrix B is also the natural logarithm of A.

A real matrix A is normal if AA^T = A^TA. For a non-singular normal matrix, if each negative eigenvalues of A occur an even number of times, A has a real logarithm. Note that this does not guarantee the uniqueness of the real logarithm.