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Pressing the On button (green) is an idempotent operation, since it has the same effect whether done once or multiple times. Likewise, pressing Off is idempotent. Idempotence ( UK : / ˌ ɪ d ɛ m ˈ p oʊ t ən s / , [ 1 ] US : / ˈ aɪ d ə m -/ ) [ 2 ] is the property of certain operations in mathematics and computer science whereby they can ...
Idempotent matrices arise frequently in regression analysis and econometrics. For example, in ordinary least squares , the regression problem is to choose a vector β of coefficient estimates so as to minimize the sum of squared residuals (mispredictions) e i : in matrix form,
A primitive idempotent of a ring R is a nonzero idempotent a such that aR is indecomposable as a right R-module; that is, such that aR is not a direct sum of two nonzero submodules. Equivalently, a is a primitive idempotent if it cannot be written as a = e + f , where e and f are nonzero orthogonal idempotents in R .
In mathematics, an idempotent binary relation is a binary relation R on a set X (a subset of Cartesian product X × X) for which the composition of relations R ∘ R is the same as R. [ 1 ] [ 2 ] This notion generalizes that of an idempotent function to relations.
In the corresponding central digraph, each idempotent vertex has a self-loop. The remaining vertices each belong to a unique 2-cycle. In the matrix view of central groupoids, the idempotent elements form the 1s on the main diagonal of a matrix representing the groupoid. Each row and column of the matrix also contains exactly 1s.
An idempotent is an element such that e 2 = e. One example of an idempotent element is a projection in linear algebra. A unit is an element a having a multiplicative inverse; in this case the inverse is unique, and is denoted by a –1.
Then the resulting idempotent monoid {lt, e, gt} models the lexicographical order of a sequence given the orders of its elements, with e representing equality. The underlying set of any ring, with addition or multiplication as the operation. (By definition, a ring has a multiplicative identity 1.)
A quasigroup with an idempotent element is called a pique ("pointed idempotent quasigroup"); this is a weaker notion than a loop but common nonetheless because, for example, given an abelian group, (A, +), taking its subtraction operation as quasigroup multiplication yields a pique (A, −) with the group identity (zero) turned into a "pointed ...