Quaternionic matrix: Difference between revisions

Revision as of 16:04, 8 September 2010

Revision as of 16:37, 8 September 2010

Line 7:

==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.

Matrix addition is defined in the usual way:

'''Addition'''. The 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>

'''Multiplication'''. The product of two quaternionic matrices ''A'' and ''B'' also follows the usual definition for [[matrix multiplication]]. ThatFor isit 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