In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, positive definiteness is a property of any object to which a
bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called '' scalars''). In other words, a bilinear form is a function that is linea ...
or a
sesquilinear form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows o ...
may be naturally associated, which is
positive-definite. See, in particular:
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Positive-definite bilinear form
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Positive-definite function
In mathematics, a positive-definite function is, depending on the context, either of two types of function.
Most common usage
A ''positive-definite function'' of a real variable ''x'' is a complex-valued function f: \mathbb \to \mathbb such ...
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Positive-definite function on a group
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Positive-definite functional
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Positive-definite kernel In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solving ...
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Positive-definite matrix
In mathematics, a symmetric matrix M with real entries is positive-definite if the real number z^\textsfMz is positive for every nonzero real column vector z, where z^\textsf is the transpose of More generally, a Hermitian matrix (that is, a ...
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Positive-definite quadratic form
References
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{{Set index article, mathematics
Quadratic forms