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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; Negative-definite function
The above rules stating that extrema are characterized (among critical points with a non-singular Hessian) by a positive-definite or negative-definite Hessian cannot apply here since a bordered Hessian can neither be negative-definite nor positive-definite, as = if is any vector whose sole non-zero entry is its first.
At the remaining critical point (0, 0) the second derivative test is insufficient, and one must use higher order tests or other tools to determine the behavior of the function at this point. (In fact, one can show that f takes both positive and negative values in small neighborhoods around (0, 0) and so this point is a saddle point of f.)
In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector , where is the row vector transpose of . [1] More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector , where denotes the ...
A function is a semivariogram if and only if it is a conditionally negative definite function, i.e. for all weights , …, subject to = = and locations , …, it holds: ∑ i = 1 N ∑ j = 1 N w i γ ( s i , s j ) w j ≤ 0 {\displaystyle \sum _{i=1}^{N}\sum _{j=1}^{N}w_{i}\gamma (\mathbf {s} _{i},\mathbf {s} _{j})w_{j}\leq 0}
If ˙ is negative definite, then the global asymptotic stability of the origin is a consequence of Lyapunov's second theorem. The invariance principle gives a criterion for asymptotic stability in the case when V ˙ ( x ) {\displaystyle {\dot {V}}(\mathbf {x} )} is only negative semidefinite.
The first method developed the solution in a series which was then proved convergent within limits. The second method, which is now referred to as the Lyapunov stability criterion or the Direct Method, makes use of a Lyapunov function V(x) which has an analogy to the potential
Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation [1] [2] is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into (usually) two matrices W and H, with the property that all three matrices have no negative elements. This non-negativity makes the resulting ...