Quaternionic matrix: Difference between revisions

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{{Short description|Concept in linear algebra}}

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A '''quaternionic matrix''' is a [[matrix (mathematics)|matrix]] whose elements are [[quaternion]]s.

==Matrix operations==

The quaternions form a [[noncommutative]] [[ring (algebra)|ring]], and therefore the also form a noncommutative ring, in which [[Matrix addition|addition]] and [[Matrix multiplication|multiplication]] can be defined for quaternionic matrices as for matrices over any ring.

'''Addition'''. The sum of two quaternionic matrices ''A'' and ''B'' is defined in the usual way by element-wise addition:

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:<math>\operatorname{trace}(AB)\ne\operatorname{trace}(BA).</math>

Left scalar multiplication, isand right scalar multiplication are defined by

:<math>(cA)_{ij}=cA_{ij}, \qquad (Ac)_{ij}=A_{ij}c.\,</math>

Again, since multiplication is not commutative some care must be taken in the order of the factors.<ref>{{cite book |title=Matrix groups for undergraduates|first=Kristopher|last=Tapp

|publisher=AMS Bookstore|year=2005|isbn=0-8218-3785-0 |pages=11 ''ff''

|url=httphttps://books.google.com/books?id=Un_15Im3NhUC&pg=PA11#v=onepage&q&f=false}}</ref>

==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|s2cid=13958298 }}</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|doi-access=free }}</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 [[square matrix|''n'' by ''n'' matrix]] is invertible exactlyif and only whenif its determinant is nonzero.

==Applications==

Quaternionic matrices are used in [[quantum mechanics]]<ref>{{cite journal |author= N. Rösch |title=Time-reversal symmetry, Kramers' degeneracy and the algebraic eigenvalue problem |year=1983 |journal=[[Chemical Physics]] |volume=80 |issue=1–2 |pages=1–5 |doi=10.1016/0301-0104(83)85163-5|bibcode=1983CP.....80....1R }}</ref> and in the treatment of [[multibody problem]]s.<ref>{{cite book |title=Quaternionic and Clifford calculus for physicists and engineers |url=https://archive.org/details/quaternionicclif00kgue |url-access=limited |author=Klaus Gürlebeck |author2=Wolfgang Sprössig |chapter=Quaternionic matrices |pages=32–34[https://archive.org/details/quaternionicclif00kgue/page/n43 32]–34 |publisher=Wiley |year=1997 |isbn=978-0-471-96200-7}}</ref>

==References==

{{reflist}}

{{Matrix classes}}

[[Category:Matrices]]

[[Category:Matrices (mathematics)]]

[[Category:Linear algebra]]