It’s possible to represent complex numbers as a pair of real numbers or 2 × 2 matrices with real entries.

z leftrightarrow (a, b) leftrightarrow begin{bmatrix}a & -b \ b & a end{bmatrix}

And it’s possible to represent quaternions as pairs of complex numbers or 2 × 2 matrices with complex entries

q leftrightarrow (z_0, z_1) leftrightarrow begin{bmatrix} z_0 & z_1 \ -z_1^* & z_0^* end{bmatrix}

were z* is the complex conjugate of z.

And it’s also possible to represent octonions as pairs of quaternions or 2 × 2 matrices with quaternion entries, with a twist.

o leftrightarrow (q_0, q_1) leftrightarrow begin{bmatrix} q_0 & q_1 \ -q_1^* & q_0^* end{bmatrix}

where q* is the quaternion conjugate of q.

Matrix multiplication is associative, but octonion multiplication is not, so something has to give. We have to change the definition of matrix multiplication slightly.

begin{bmatrix} alpha_0 & alpha_1 \ alpha_2 & alpha_3 end{bmatrix}circbegin{bmatrix} beta_0 & beta_1 \ beta_2 & beta_3 end{bmatrix}=begin{bmatrix} alpha_0beta_0+beta_2alpha_1 & beta_1alpha_0+alpha_1beta_3\ beta_0alpha_2+alpha_3beta_2 & alpha_2beta_1+alpha_3beta_3 end{bmatrix}

In half the products, the beta term comes before the alpha term. This wouldn’t matter if the alpha and beta terms commuted, e.g. if they were complex numbers this this would be ordinary matrix multiplication. But the alphas and betas are quaternions, and so order matters, and the matrix product defined above is not the standard matrix product.

Going back to the idea of matrices of matrices that I wrote about a few days ago, we could represent the octonions as 2 × 2 matrices whose entries are 2 × 2 matrices of complex numbers, etc.

If you look closely at the matrix representations above, you’ll notice that the matrix representations of quaternions and octonions doesn’t quite match the pattern of the complex numbers. There should be a minus sign in the top right corner and not in the bottom left corner. You could do it that way, but there’s a sort of clash of conventions going on here.

The post Representing octonions as matrices, sorta first appeared on John D. Cook.

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