Quaternionic matrix: Difference between revisions

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== Product of quaternionic matrices ==

The product of two quaternionic matrices is eitherfollows ''hamiltonian''the orusual ''octonionic'',definition dependingfor on[[matrix whethermultiplication]]. theThat orderis, ofthe multiplicationentry ofin the entries''i''th isrow preserved.and The''j''th quaternioncolumn field,of <math>\mathbb{H}</math>,the product is non-commutative,the so the[[dot product]] of twothe quaternionic''i''th matricesrow mayof orthe mayfirst notmatrix preservewith the order''j''th column of multiplicationthe second matrix. Specifically:

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

For example, for

=== Hamiltonian product ===

The hamiltonian product preserves the order of multiplication.

:<math>

U =

Line 19 ⟶ 16:

u_{11} & u_{12}\\

u_{21} & u_{22}\\

\end{pmatrix},

\quad

V =

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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.<ref>Tapp pp. 11 ff. for the section.</ref>

==References==