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
Famous quotes containing the words relationship with and/or relationship:
“Christianity as an organized religion has not always had a harmonious relationship with the family. Unlike Judaism, it kept almost no rituals that took place in private homes. The esteem that monasticism and priestly celibacy enjoyed implied a denigration of marriage and parenthood.”
—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)
“We think of religion as the symbolic expression of our highest moral ideals; we think of magic as a crude aggregate of superstitions. Religious belief seems to become mere superstitious credulity if we admit any relationship with magic. On the other hand our anthropological and ethnographical material makes it extremely difficult to separate the two fields.”
—Ernst Cassirer (1874–1945)