Geometric Algebra - Relationship With Other Formalisms

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

 Z = {e_1} P = {e_1} ( x {e_1} + y {e_2})
= x (1) + y ( {e_1} {e_2})\,

where we identify ie1e2 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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