Quaternionic matrix

A quaternionic matrix is a matrix whose elements are quaternions.

Product of quaternionic matrices

The product of two quaternionic matrices follows the usual definition for matrix multiplication. That is, the entry in the ith row and jth column of the product is the dot product of the ith row of the first matrix with the jth column of the second matrix. Specifically:

$(AB)_{ij}=\sum _{s}A_{is}B_{sj}.\,$

For example, for

$U={\begin{pmatrix}u_{11}&u_{12}\\u_{21}&u_{22}\\\end{pmatrix}},\quad V={\begin{pmatrix}v_{11}&v_{12}\\v_{21}&v_{22}\\\end{pmatrix}},$

the product is

$UV={\begin{pmatrix}u_{11}v_{11}+u_{12}v_{21}&u_{11}v_{12}+u_{12}v_{22}\\u_{21}v_{11}+u_{22}v_{21}&u_{21}v_{12}+u_{22}v_{22}\\\end{pmatrix}}.$

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

References

• Tapp, Kristopher (2005). Matrix groups for undergraduates. AMS Bookstore. ISBN 0821837850.