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
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.
Read more about this topic: Quaternion
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... Then the complex matrix , with (complex conjugates of u and v), represents q in the ring of matrices in the sense that multiplication of split-quaternions behaves the same way as the ... For example, the determinant of this matrix is The appearance of the minus sign, where there is a plus in H, distinguishes coquaternions from quaternions ... Besides the complex matrix representation, another linear representation associates coquaternions with 2 × 2 real matrices ...
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