Quaternionic matrix: Difference between revisions

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==Determinants==

There is no natural way to define a [[determinant]] for (square) quaternionic matrices so that the values of the determinant are quaternions.<ref>{{cite journal |author=Helmer Aslaksen |title=Quaternionic determinants |year=19661996 |journal=[[The Mathematical Intelligencer]] |volume=18 |number=3 |pages=57–65 |doi=10.1007/BF03024312}}</ref> Complex valued determinants can be defined however.<ref>{{cite journal |author=E. Study |title=Zur Theorie der linearen Gleichungen |year=1920 |journal=[[Acta Mathematica]] |volume=42 |number=1 |pages=1–61 |language=[[German language|German]] |doi=10.1007/BF02404401}}</ref> The quaternion ''a'' + ''bi'' + ''cj'' + ''dk'' can be represented as the 2&times;2 complex matrix

: <math>\begin{bmatrix}~~a+bi & c+di \\ -c+di & a-bi \end{bmatrix}.</math>

This defines a map Ψ_''mn'' from the ''m'' by ''n'' quaternionic matrices to the 2''m'' by 2''n'' complex matrices by replacing each entry in the quaternionic matrix by its 2 by 2 complex representation. The complex valued determinant of a square quaternionic matrix ''A'' is then defined as det(Ψ(''A'')). Many of the usual laws for determinants hold; in particular, an ''n'' by ''n'' matrix is invertible exactly when its determinant is nonzero.