positive definite

Adjective

Meaning

positive definite (not comparable)

Alternative form of positive-definite

(linear algebra) Said of a real, square matrix: that the product of it with a column vector on its right side and the transpose of that column vector on its left side is greater or equal to zero, only equaling zero if the vector itself is zero.

(linear algebra) Said of a quadratic form: that the application of it to a tuple is greater or equal to zero, only equaling zero if the tuple itself is zero.

Alternative names

positive-definite

Antonyms

Examples

Further properties If M is a Hermitian positive-semidefinite matrix, one sometimes writes M ≥ 0 and if M is positive-definite one writes M > 0. This may be confusing, as sometimes nonnegative matrices are also denoted in this way. Source: Internet

When Σ is positive-definite, the Cholesky decomposition is typically used, and the extended form of this decomposition can always be used (as the covariance matrix may be only positive semi-definite) in both cases a suitable matrix A is obtained. Source: Internet

So, an inner product on a real vector space is a positive-definite symmetric bilinear form. Source: Internet

Inner Product Spaces may be defined over any field, having "inner products" that are linear in the first(by convention) argument, conjugate-symmetrical, and positive-definite. Source: Internet

The cross product of two vectors in 3 dimensions with positive-definite quadratic form is closely related to their outer product. Source: Internet

Unlike Inner Products, Scalar Products and Hermitian Products need not be positive-definite. Source: Internet

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