Quaternionic matrix

A quaternionic matrix is a matrix whose elements are quaternions.

Product of quaternionic matrices

The product of two quaternionic matrices is either hamiltonian or octonionic, depending on whether the order of multiplication of the entries is preserved. The quaternion field, $\mathbb {H}$, is non-commutative, 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 secondary 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

• Quaternion
• Hamilton
• Octonion