Peirce Decomposition
If e is an idempotent in a Jordan algebra A (e2=e) and R is the operation of multiplication by e, then
- R(2R−1)(R−1) = 0
so the only eigenvalues of R are 0, 1/2, 1. If the Jordan algebra A is finite-dimensional over a field of characteristic not 2, this implies that it is a direct sum of subspaces A = A0(e) ⊕ A1/2(e) ⊕ A1(e) of the three eigenspaces. This decomposition was introduced by Albert (1947) and is called the Peirce decomposition of A relative to the idempotent e.
Read more about this topic: Jordan Algebra
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“Generality is, indeed, an indispensable ingredient of reality; for mere individual existence or actuality without any regularity whatever is a nullity. Chaos is pure nothing.”
—Charles Sanders Peirce (1839–1914)