Quaternionic matrix

Definition

One talk about quaternionic matrix when the elements are quaternions.

Product of two quaternionic matrix

The product of two quaternionic matrix is or hamiltonian or octonionic. The field $\mathbb {H}$ is a non-commutative field , so the product of two quaternionic matrix has or not to preserve the order of the multiplication .

Hamiltonian product

The hamiltonian product preserves the order of the multiplication. $U={\begin{pmatrix}u_{11}&u_{12}\\u_{21}&u_{22}\\\end{pmatrix}}\quad V={\begin{pmatrix}v_{11}&v_{12}\\v_{21}&v_{22}\\\end{pmatrix}}\quad UV={\begin{pmatrix}u_{11}v_{11}+u_{12}v_{21}&u_{11}v_{12}+u_{12}v_{22}\\u_{21}v_{11}+u_{22}v_{21}&u_{21}v_{12}+u_{22}v_{22}\\\end{pmatrix}}$

Octonionic product

The octonionic product does not preserve the order of the multiplication : on the main diagonal , there is an inversion of the second product and on the second diagonal there is an inversion of the first products. $U={\begin{pmatrix}u_{11}&u_{12}\\u_{21}&u_{22}\\\end{pmatrix}}\quad V={\begin{pmatrix}v_{11}&v_{12}\\v_{21}&v_{22}\\\end{pmatrix}}\quad UV={\begin{pmatrix}u_{11}v_{11}+v_{21}u_{12}&v_{12}u_{11}+u_{12}v_{22}\\v_{11}u_{21}+u_{22}v_{21}&u_{21}v_{12}+v_{22}u_{22}\\\end{pmatrix}}$

See also

• matrix
• Quaternion
• Hamilton
• Octonion