Orthogonal idempotent
= Orthogonal idempotent
{title2=$e_i^2=e_i,\quad e_ie_j=e_je_i=0\ (i\ne j)$}
Distinct <idempotents> are orthogonal when their products in both orders vanish. A finite family summing to one gives a decomposition $M=\bigoplus_ie_iM$ of every left <module> as a vector space; the summands need not be submodules unless the idempotents are central. Vertex idempotents of a <path algebra> supply the spaces of its <quiver representation>.