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by Ciro Santilli (@cirosantilli, 37)

Existence of the matrix logarithm

 ... Function by signature Numeric function Exponentiation Exponential function Matrix exponential Logarithm of a matrix
 0 By others on same topic  0 Discussions  Updated 2025-05-29  +Created 1970-01-01  See my version
en.wikipedia.org/wiki/Logarithm_of_a_matrix#Existence mentions it always exists for all invertible complex matrices. But the real condition is more complicated. Notable counter example: -1 cannot be reached by any real etk.
The Lie algebra exponential covering problem can be seen as a generalized version of this problem, because
  • Lie algebra of GL(n) is just the entire Mn​
  • we can immediately exclude non-invertible matrices from being the result of the exponential, because etM has inverse e−tM, so we already know that non-invertible matrices are not reachable

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  1. Logarithm of a matrix
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  • cirosantilli/the-matrix-exponential-is-not-surjective

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