Quaternionic matrix: Difference between revisions

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{{Short description|Concept in linear algebra}}

{{orphan|date=January 2010}}

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

==Matrix operations==

A '''quaternionic matrix''' is a [[matrix]] whose elements are [[quaternion]]s.

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

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

:<math>(A+B)_{ij}=A_{ij}+B_{ij}.\,</math>

The product of two quaternionic matrices is either or hamiltonian or octonionic. The field <math>\mathbb{H}</math> is a non-commutative field, so the product of two quaternionic matrices may or may not preserve the order of multiplication.

=== Hamiltonian product ===

The hamiltonian product preserves the order of multiplication.

'''Multiplication'''. The product of two quaternionic matrices ''A'' and ''B'' also follows the usual definition for matrix multiplication. For it to be defined, the number of columns of ''A'' must equal the number of rows of ''B''. Then the entry in the ''i''th row and ''j''th column of the product is the [[dot product]] of the ''i''th row of the first matrix with the ''j''th column of the second matrix. Specifically:

:<math>(AB)_{ij}=\sum_s A_{is}B_{sj}.\,</math>

For example, for

:<math>

U =

Line 16:

u_{11} & u_{12}\\

u_{21} & u_{22}\\

\end{pmatrix},

\quad

V =

Line 22:

v_{11} & v_{12}\\

v_{21} & v_{22}\\

\end{pmatrix},

</math>

\quad

the product is

:<math>

UV =

\begin{pmatrix}

u_{11}v_{11}+u_{12}v_{21} & u_{11}v_{12}+u_{12}v_{22}\\

u_{21}v_{11}+u_{22}v_{21} & u_{21}v_{12}+u_{22}v_{22}\\

\end{pmatrix}.

</math>

Since quaternionic multiplication is noncommutative, care must be taken to preserve the order of the factors when computing the product of matrices.

The [[Identity element|identity]] for this multiplication is, as expected, the diagonal matrix I&nbsp;=&nbsp;diag(1, 1, ... , 1). Multiplication follows the usual laws of [[associativity]] and [[distributivity]]. The trace of a matrix is defined as the sum of the diagonal elements, but in general

=== Octonionic product ===

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

Left scalar multiplication, and right scalar multiplication are defined by

The octonionic product does not preserve the order of multiplication: on the main diagonal, there is an inversion of the second product and on the second diagonal there is an inversion of the first product.

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

<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

U =

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

\begin{pmatrix}

|url=https://books.google.com/books?id=Un_15Im3NhUC&pg=PA11}}</ref>

u_{11} & u_{12}\\

u_{21} & u_{22}\\

\end{pmatrix}

\quad

==Determinants==

V =

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=1996 |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 |doi=10.1007/BF02404401|doi-access=free }}</ref> The quaternion ''a'' + ''bi'' + ''cj'' + ''dk'' can be represented as the 2&times;2 complex matrix

\begin{pmatrix}

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

v_{11} & v_{12}\\

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 if and only if its determinant is nonzero.

v_{21} & v_{22}\\

\end{pmatrix}

\quad

==Applications==

UV =

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=[https://archive.org/details/quaternionicclif00kgue/page/n43 32]–34 |publisher=Wiley |year=1997 |isbn=978-0-471-96200-7}}</ref>

\begin{pmatrix}

u_{11}v_{11} + v_{21}u_{12} & v_{12}u_{11} + u_{12}v_{22}\\

==References==

v_{11}u_{21} + u_{22}v_{21} & u_{21}v_{12} + v_{22}u_{22}\\

{{reflist}}

\end{pmatrix}

</math>

{{Matrix classes}}

== See also ==

*[[Matrix]]

*[[Quaternion]]

*[[William Rowan Hamilton|Hamilton]]

*[[Octonion]]

[[Category:CalculationMatrices by Matrix(mathematics)]]

[[Category:Linear algebra]]