# Quaternion - Matrix Representations

Matrix Representations

Just as complex numbers can be represented as matrices, so can quaternions. There are at least two ways of representing quaternions as matrices in such a way that quaternion addition and multiplication correspond to matrix addition and matrix multiplication. One is to use 2×2 complex matrices, and the other is to use 4×4 real matrices. In each case, the representation given is one of a family of linearly related representations. In the terminology of abstract algebra, these are injective homomorphisms from H to the matrix rings M2(C) and M4(R), respectively.

Using 2×2 complex matrices, the quaternion a + bi + cj + dk can be represented as

This representation has the following properties:

• Complex numbers (c = d = 0) correspond to diagonal matrices.
• The norm of a quaternion (the square root of a product with its conjugate, as with complex numbers) is the square root of the determinant of the corresponding matrix.
• The conjugate of a quaternion corresponds to the conjugate transpose of the matrix.
• Restricted to unit quaternions, this representation provides an isomorphism between S3 and SU(2). The latter group is important for describing spin in quantum mechanics; see Pauli matrices.

Using 4×4 real matrices, that same quaternion can be written as

$begin{bmatrix} a & b & c & d \ -b & a & -d & c \ -c & d & a & -b \ -d & -c & b & a end{bmatrix}$
$= a begin{bmatrix} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 end{bmatrix} + b begin{bmatrix} 0 & 1 & 0 & 0 \ -1 & 0 & 0 & 0 \ 0 & 0 & 0 & -1 \ 0 & 0 & 1 & 0 end{bmatrix} + c begin{bmatrix} 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \ -1 & 0 & 0 & 0 \ 0 & -1 & 0 & 0 end{bmatrix} + d begin{bmatrix} 0 & 0 & 0 & 1 \ 0 & 0 & -1 & 0 \ 0 & 1 & 0 & 0 \ -1 & 0 & 0 & 0 end{bmatrix}.$

In this representation, the conjugate of a quaternion corresponds to the transpose of the matrix. The fourth power of the norm of a quaternion is the determinant of the corresponding matrix. Complex numbers are block diagonal matrices with two 2×2 blocks.