Wireless Ad Hoc and Sensor Networks

56 Wireless Ad Hoc and Sensor NetworksIf Q is symmetric, then it is positive definite if and only if all its eigenvaluesare positive and positive semidefinite if and only if all its eigenvaluesare nonnegative. If Q is not symmetric, tests are more complicatedand involve determining the minors of the matrix. Tests for negativedefiniteness and semidefiniteness may be found by noting that Q is negative(semi) definite if and only if Q is positive (semi) definite.If Q is a symmetric matrix, its singular values are the magnitudes of itseigenvalues. If Q is a symmetric positive semidefinite matrix, its singularvalues and its eigenvalues are the same. If Q is positive semidefinite then,for any vector x , one has the useful inequalityσmin2( Q)| x| x T2≤ Qx≤σ( Q)| x|max(2.20)2.2.2 Continuity and Function NormsnmGiven a subset S ⊂R , a function f( x): S→R is continuous on x0 ∈Sif,for every ε > 0, there exists a δε ( , x 0 ) > 0 such that | x− x0| < δε ( , x0)impliesthat |( f x) − f( x0)| < ε.If δ is independent of x 0 , then the function is said tobe uniformly continuous. Uniform continuity is often difficult to test.However, if f( x) is continuous and its derivative f′( x)is bounded, then itis uniformly continuous.n mA function f( x):R →R is differentiable if its derivative f′( x)exists.It is continuously differentiable, if its derivative exists and is continuous.nf( x) is said to be locally Lipschitz if, for all x, z∈S⊂R, one has|( f x) − f()| z < L| x−z|(2.21)for some finite constant LS ( ), where L is known as a Lipschitz constant.nIf S =R , then the function is globally Lipschitz.If f( x)is globally Lipschitz, then it is uniformly continuous. If it iscontinuously differentiable, it is locally Lipschitz. If it is differentiable, itis continuous. For example, f( x)= x2 is continuously differentiable. It islocally, but not globally, Lipschitz. It is continuous but not uniformlycontinuous.nGiven a function ft ():[ 0, ∞→R ) , according to Barbalat’s Lemma, if∞∫ 0ftdt () ≤∞(2.22)and ft () is uniformly continuous, then ft ()→ 0 as t →∞.