**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 M_{2}(**C**) and M_{4}(**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
*S*3 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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### Famous quotes containing the word matrix:

“As all historians know, the past is a great darkness, and filled with echoes. Voices may reach us from it; but what they say to us is imbued with the obscurity of the *matrix* out of which they come; and try as we may, we cannot always decipher them precisely in the clearer light of our day.”

—Margaret Atwood (b. 1939)