In mathematics, a canonical basis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context:
In a coordinate space, and more generally in a free module, it refers to the standard basis defined by the Kronecker delta.
In a polynomial ring, it refers to its standard basis given by the monomials, .
For finite extension fields, it means the polynomial basis.
In linear algebra, it refers to a set of n linearly independent generalized eigenvectors of an n×n matrix , if the set is composed entirely of Jordan chains.[1]
In representation theory, it refers to the basis of the quantum groups introduced by Lusztig.
In mathematics, a canonicalbasis is a basis of an algebraic structure that is canonical in a sense that depends on the precise context: In a coordinate...
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF), is an upper triangular matrix of a particular form called a Jordan...
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical...
canonicalbasis if it is composed entirely of Jordan chains. Thus, once we have determined that a generalized eigenvector of rank m is in a canonical...
In mathematics, the standard basis (also called natural basis or canonicalbasis) of a coordinate vector space (such as R n {\displaystyle \mathbb {R}...
The CanonicalBasis of the High Council (Albanian: Bazet e Kanunores së Këshillës së Naltë), often referred to as the "Statute of Lushnje", was the first...
In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium...
explicit form of the Hamiltonian itself. Canonical transformations are useful in their own right, and also form the basis for the Hamilton–Jacobi equations (a...
A} is an n × n matrix whose columns, considered as vectors, form a canonicalbasis for A {\displaystyle A} and appear in M {\displaystyle M} according...
be viewed as specializations as v → 0 {\displaystyle v\to 0} of the canonicalbasis defined by Lusztig (1990). As a consequence of its defining relations...
particular spanning set is also a basis. If (−1, 0, 0) were replaced by (1, 0, 0), it would also form the canonicalbasis of R 3 {\displaystyle \mathbb {R}...
the Jewish Tanakh as the textual basis for their translations of the protocanonical books (those accepted as canonical by both Jews and all Christians)...
the Deuterocanon (DC), are certain books and passages considered to be canonical books of the Old Testament by the Catholic Church, the Eastern Orthodox...
the Frobenius normal form or rational canonical form of a square matrix A with entries in a field F is a canonical form for matrices obtained by conjugation...
In the practice of Christianity, canonical hours mark the divisions of the day in terms of fixed times of prayer at regular intervals. A book of hours...
Pn(K) = P(Kn+1) is considered. It has a canonical frame consisting of the image by p of the canonicalbasis of Kn+1 (consisting of the elements having...
In statistical mechanics, the grand canonical ensemble (also known as the macrocanonical ensemble) is the statistical ensemble that is used to represent...
suffices to construct the morphism that sends the elements of the canonicalbasis of Rn to the generators of the module, and take the quotient by its...
^{-1}}}\ V.} If V = Rn is endowed with its canonicalbasis e1, …, en, and if the matrix of T in this basis is A, then the adjugate of T is the adjugate...
In statistics, canonical-correlation analysis (CCA), also called canonical variates analysis, is a way of inferring information from cross-covariance...
In mathematics, the canonical bundle of a non-singular algebraic variety V {\displaystyle V} of dimension n {\displaystyle n} over a field is the line...
+){\displaystyle (+,+)} on R2{\displaystyle \mathbb {R} ^{2}} (with the canonicalbasis) is given by: R2×R2→R2,(x1y1)⋅(x2y2)=x1x2+y1y2{\displaystyle \mathbb...
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