Quaternionic matrix: Difference between revisions

Revision as of 16:12, 2 September 2010

Revision as of 16:36, 2 September 2010

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

==Matrix operations==

== Product of quaternionic matrices ==

Matrix addition is defined in the usual way:

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

The product of two quaternionic matrices follows the usual definition for [[matrix multiplication]]. That is, 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

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</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>

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

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

==References==