Editing Pauli matrices (section)

=== Quaternions ===
{{main|Spinor#Three dimensions}}
The real linear span of {{math|{''I'', ''iσ''{{sub|1}}, ''iσ''{{sub|2}}, ''iσ''{{sub|3}}<nowiki>}</nowiki>}} is isomorphic to the real algebra of [[quaternions]], <math>\mathbb{H}</math>, represented by the span of the basis vectors <math> \left\{\; \mathbf{1}, \, \mathbf{i}, \, \mathbf{j}, \, \mathbf{k} \;\right\} .</math> The isomorphism from <math>\mathbb{H}</math> to this set is given by the following map (notice the reversed signs for the Pauli matrices):
<math display=block>
  \mathbf{1} \mapsto I, \quad
  \mathbf{i} \mapsto - \sigma_2\sigma_3 = - i\,\sigma_1, \quad
  \mathbf{j} \mapsto - \sigma_3\sigma_1 = - i\,\sigma_2, \quad
  \mathbf{k} \mapsto - \sigma_1\sigma_2 = - i\,\sigma_3.
</math>

Alternatively, the isomorphism can be achieved by a map using the Pauli matrices in reversed order,<ref>
{{cite book
 |first=Mikio |last=Nakahara
 |year=2003
 |title=Geometry, Topology, and Physics
 |edition=2nd  |page=[https://books.google.com/books?id=cH-XQB0Ex5wC&pg=PR22 xxii]
 |publisher=CRC Press
 |isbn=978-0-7503-0606-5
 |url=https://books.google.com/books?id=cH-XQB0Ex5wC&q=%22Pauli+matrices%22+OR+%22Pauli+matrix%22
 |via=Google Books
}}
</ref>
:<math>
  \mathbf{1} \mapsto I, \quad
  \mathbf{i} \mapsto i\,\sigma_3 \, , \quad
  \mathbf{j} \mapsto i\,\sigma_2 \, , \quad
  \mathbf{k} \mapsto i\,\sigma_1 ~ .
</math>

As the set of [[versor]]s {{math|''U'' ⊂ <math>\mathbb{H}</math>}} forms a group isomorphic to {{math|SU(2)}}, {{mvar|U}} gives yet another way of describing {{math|SU(2)}}. The two-to-one homomorphism from {{math|SU(2)}} to {{math|SO(3)}} may be given in terms of the Pauli matrices in this formulation.