Quaternionic matrix

A quaternionic matrix is a matrix whose elements are quaternions.

Product of two quaternionic matrices

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

Hamiltonian product

The hamiltonian product preserves the order of 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 multiplication: on the main diagonal, there is an inversion of the second product and on the reverse diagonal there is an inversion of the first product.

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

See also

• Matrix
• Quaternion
• Hamilton
• Octonion