A quaternionic matrix is a matrix whose elements are quaternions.
Matrix operations The quaternions form a noncommutative ring, and therefore addition and multiplication can be defined for quaternionic matrices as for matrices over any ring. Addition. The sum of two quaternionic matrices A and B is defined in the usual way by element-wise addition:
( A + B ) i j = A i j + B i j . {\displaystyle (A+B)_{ij}=A_{ij}+B_{ij}.\,}
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 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:
( A B ) i j = ∑ s A i s B s j . {\displaystyle (AB)_{ij}=\sum _{s}A_{is}B_{sj}.\,}
For example, for
U = ( u 11 u 12 u 21 u 22 ) , V = ( v 11 v 12 v 21 v 22 ) , {\displaystyle 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
U V = ( 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 ) . {\displaystyle 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. The identity for this multiplication is, as expected, the diagonal matrix I = 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
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