Logarithm of a matrix: Difference between revisions

Revision as of 04:04, 18 September 2022

Revision as of 11:28, 18 September 2022

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==Constraints in the 2 × 2 case==

If a 2 × 2 real matrix has a negative [[determinant]], it has no real logarithm. Note first that any 2 × 2 real matrix can be considered one of the three types of the complex[[hypercomplex number]] ''z'' = ''x'' + ''y'' ε, where ε² ∈ { −1, 0, +1 }. This ''z'' is a point on a complex subplane of the [[ring (mathematics)|ring]] of matrices.<ref>{{Wikibooks-inline|Abstract Algebra/2x2 real matrices}}</ref>

The case where the determinant is negative only arises in a plane with ε² =+1, that is a [[split-complex number]] plane. Only one quarter of this plane is the image of the exponential map, so the logarithm is only defined on that quarter (quadrant). The other three quadrants are images of this one under the [[Klein four-group]] generated by ε and −1.

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\begin{pmatrix}1.06 & .35\\ .35 & 1.06 \end{pmatrix} ~. </math>

For a richer example, start with a [[pythagoreanPythagorean triple]] (''p,q,r'')

and let {{math|''a'' {{=}} log(''p'' + ''r'') &minus; log ''q''}}. Then

:<math>e^a = \frac {p + r} {q} = \cosh a + \sinh a</math>.