Relationship With Other Formalisms
may be directly compared to vector algebra.
The even subalgebra of is isomorphic to the complex numbers, as may be seen by writing a vector P in terms of its components in an orthonormal basis and left multiplying by the basis vector e1, yielding
where we identify i ↦ e1e2 since
Similarly, the even subalgebra of with basis {1, e2e3, e3e1, e1e2} is isomorphic to the quaternions as may be seen by identifying i ↦ −e2e3, j ↦ −e3e1 and k ↦ −e1e2.
Every associative algebra has a matrix representation; the Pauli matrices are a representation of and the Dirac matrices are a representation of, showing the equivalence with matrix representations used by physicists.
Read more about this topic: Geometric Algebra
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—Beatrice Gottlieb, U.S. historian. The Family in the Western World from the Black Death to the Industrial Age, ch. 12, Oxford University Press (1993)
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