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Positive-definiteness arises naturally in the theory of the Fourier transform; it can be seen directly that to be positive-definite it is sufficient for f to be the Fourier transform of a function g on the real line with g(y) ≥ 0.
A Lyapunov function for an autonomous dynamical system {: ˙ = ()with an equilibrium point at = is a scalar function: that is continuous, has continuous first derivatives, is strictly positive for , and for which the time derivative ˙ = is non positive (these conditions are required on some region containing the origin).
In particular, the discrete-time Lyapunov equation (also known as Stein equation) for is A X A H − X + Q = 0 {\displaystyle AXA^{H}-X+Q=0} where Q {\displaystyle Q} is a Hermitian matrix and A H {\displaystyle A^{H}} is the conjugate transpose of A {\displaystyle A} , while the continuous-time Lyapunov equation is
Positive-definite functions on are intimately related to unitary representations of . Every unitary representation of G {\displaystyle G} gives rise to a family of positive-definite functions. Conversely, given a positive-definite function, one can define a unitary representation of G {\displaystyle G} in a natural way.
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 integral operator equations. Since then, positive-definite functions and their various analogues ...
In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular: Positive-definite bilinear form; Positive-definite function; Positive-definite function on a group; Positive-definite functional; Positive-definite kernel
is assumed to be positive-definite, and the vectors and must satisfy the curvature condition s k T y k = s k T B s k > 0. {\displaystyle s_{k}^{T}y_{k}=s_{k}^{T}Bs_{k}>0.} The DFP formula is quite effective, but it was soon superseded by the Broyden–Fletcher–Goldfarb–Shanno formula , which is its dual (interchanging the roles of y and s ).
This makes a positive definite matrix. More properties of controllable systems can be found in Chen (1999 , p. 145 ), as well as the proof for the other equivalent statements of “The pair ( A , B ) {\displaystyle ({\boldsymbol {A}},{\boldsymbol {B}})} is controllable” presented in section Controllability in LTI Systems.