Existence
The question of whether a matrix has a logarithm has the easiest answer when considered in the complex setting. A matrix has a logarithm if and only if it is invertible. The logarithm is not unique, but if a matrix has no negative real eigenvalues, then it has a unique logarithm whose eigenvalues lie all in the strip {z ∈ C | −π < Im z < π}. This logarithm is known as the principal logarithm.
The answer is more involved in the real setting. A real matrix has a real logarithm if and only if it is invertible and each Jordan block belonging to a negative eigenvalue occurs an even number of times. If an invertible real matrix does not satisfy the condition with the Jordan blocks, then it has only complex logarithms. This can already be seen in the scalar case: the logarithm of −1 is a complex number. The existence of real matrix logarithms of real 2 x 2 matrices is considered in a later section.
Read more about this topic: Logarithm Of A Matrix
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—George Bernard Shaw (1856–1950)