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Negative-definite Bilinear Form
In mathematics, negative definiteness is a property of any object to which a bilinear form may be naturally associated, which is negative-definite. See, in particular: * Negative-definite bilinear form * Negative-definite quadratic form * Negative-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 c ... * Negative-definite function {{Set index article, mathematics Quadratic forms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 linear in each argument separately: * and * and The dot product on \R^n is an example of a bilinear form. The definition of a bilinear form can be extended to include modules over a ring, with linear maps replaced by module homomorphisms. When is the field of complex numbers , one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument. Coordinate representation Let be an -dimensional vector space with basis . The matrix ''A'', defined by is called the ''matrix of the bilinear form'' on the basis . If the matrix represents a vector with respect to this basis, and analogously, represents another vector , then: B(\mathbf, \mathbf) = \mathbf^\textsf A\mathbf = \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negative-definite Bilinear Form
In mathematics, negative definiteness is a property of any object to which a bilinear form may be naturally associated, which is negative-definite. See, in particular: * Negative-definite bilinear form * Negative-definite quadratic form * Negative-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 c ... * Negative-definite function {{Set index article, mathematics Quadratic forms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negative-definite Bilinear Form
In mathematics, negative definiteness is a property of any object to which a bilinear form may be naturally associated, which is negative-definite. See, in particular: * Negative-definite bilinear form * Negative-definite quadratic form * Negative-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 c ... * Negative-definite function {{Set index article, mathematics Quadratic forms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negative-definite Quadratic Form
In mathematics, negative definiteness is a property of any object to which a bilinear form may be naturally associated, which is negative-definite. See, in particular: * Negative-definite bilinear form * Negative-definite quadratic form * Negative-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 c ... * Negative-definite function {{Set index article, mathematics Quadratic forms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negative-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 complex matrix equal to its conjugate transpose) is positive-definite if the real number z^* Mz is positive for every nonzero complex column vector z, where z^* denotes the conjugate transpose of z. Positive semi-definite matrices are defined similarly, except that the scalars z^\textsfMz and z^* Mz are required to be positive ''or zero'' (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called indefinite. A matrix is thus positive-definite if and only if it is the matrix of a positive-definite quadratic form or Hermitian form. In other words, a matrix is positive-definite if and only if it defines a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negative-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 that for any real numbers ''x''1, …, ''x''''n'' the ''n'' × ''n'' matrix : A = \left(a_\right)_^n~, \quad a_ = f(x_i - x_j) is positive ''semi-''definite (which requires ''A'' to be Hermitian; therefore ''f''(−''x'') is the complex conjugate of ''f''(''x'')). In particular, it is necessary (but not sufficient) that : f(0) \geq 0~, \quad , f(x), \leq f(0) (these inequalities follow from the condition for ''n'' = 1, 2.) A function is ''negative semi-definite'' if the inequality is reversed. A function is ''definite'' if the weak inequality is replaced with a strong ( 0). Examples If (X, \langle \cdot, \cdot \rangle) is a real inner product space, then g_y \colon X \to \mathbb, x \mapsto \exp(i \langle y, x \rangle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
