Editing Matrix exponential (section)

== Matrix-matrix exponentials ==
The matrix exponential of another matrix (matrix-matrix exponential),<ref>{{cite web |url=http://www.rockefeller.edu/labheads/cohenje/PDFs/215BarrabasCohenalApp19941.pdf |author=Ignacio Barradas and Joel E. Cohen |date=1994 |title=Iterated Exponentiation, Matrix-Matrix Exponentiation, and Entropy |publisher=Academic Press, Inc. |url-status=dead |archive-url=https://web.archive.org/web/20090626134412/http://www.rockefeller.edu/labheads/cohenje/PDFs/215BarrabasCohenalApp19941.pdf |archive-date=2009-06-26 }}</ref> is defined as
<math display="block">X^Y = e^{\log(X) \cdot Y}</math>
<math display="block">^Y\!X = e^{Y \cdot \log(X)}</math>
for any [[normal matrix|normal]] and [[Algebraic curve#Singularities|non-singular]] {{math|''n'' × ''n''}} matrix {{mvar|X}}, and any complex {{math|''n'' × ''n''}} matrix {{mvar|Y}}.

For matrix-matrix exponentials, there is a distinction between the left exponential {{mvar|<sup>Y</sup>X}} and the right exponential {{mvar|X<sup>Y</sup>}}, because the multiplication operator for matrix-to-matrix is not [[commutative]]. Moreover,
* If {{mvar|X}} is normal and non-singular, then {{mvar|X<sup>Y</sup>}} and {{mvar|<sup>Y</sup>X}} have the same set of eigenvalues.
* If {{mvar|X}} is normal and non-singular, {{mvar|Y}} is normal, and {{math|1=''XY'' = ''YX''}}, then {{math|1=''X<sup>Y</sup>'' = ''<sup>Y</sup>X''}}.
* If {{mvar|X}} is normal and non-singular, and {{mvar|X}}, {{mvar|Y}}, {{mvar|Z}} commute with each other, then {{math|1=''X''<sup>''Y''+''Z''</sup> = ''X<sup>Y</sup>''·''X<sup>Z</sup>''}} and {{math|1=<sup>''Y''+''Z''</sup>''X'' = ''<sup>Y</sup>X''·''<sup>Z</sup>X''}}.