Wireless Ad Hoc and Sensor Networks

Background 55with tr(.) the matrix trace (i.e., sum of diagonal elements). Though theFrobenius norm is not an induced norm, it is compatible with the vector2-norm so that| Ax| ≤ | A| | x|2 F 2(2.15a)2.2.1.1 Singular Value DecompositionThe matrix norm | A|2 induced by the vector 2-norm is the maximumsingular value of A. For a general m×n matrix A, one may write thesingular value decomposition (SVD)A= UΣV Twhere U is m× n, V is n× n,and both are orthogonal; that is,(2.16)TTUU= UU = ImTTVV= VV = In(2.17)where I n is the n× n identity matrix. The m×n singular value matrix hasthe structureΣ=diag{ σ1, σ2, …, σr, 0, …, 0}(2.18)where r is the rank of A and σ i are the singular values of A. It is conventionalto arrange the singular values in a nonincreasing order, so that thelargest singular value is σmax( A ) = σ 1 . If A is full rank, then r is equal toeither m or n, whichever is smaller. Then the minimum singular valueis σmin( A)= σr(otherwise, the minimum singular value is equal to zero).The SVD generalizes the notion of eigenvalues to general nonsquarematrices. The singular values of A are the (positive) square roots of thenonzero eigenvalues of AA T , or equivalently AAT .2.2.1.2 Quadratic Forms and DefinitenessTGiven an n× n matrix, Q the quadratic form xQx, with x being an n-vector,will be important for stability analysis in this book. The quadratic formcan, in some cases, have certain properties that are independent of thevector x selected. Four important definitions are:TQ is positive definite, denoted Q > 0, if xQx> 0, ∀x ≠0.TQ is positive semidefinite, denoted Q ≥ 0, if xQx≥0, ∀x.TQ is negative definite, denoted Q < 0, if xQx< 0, ∀x ≠0.TQ is negative semidefinite, denoted Q ≤ 0, if xQx≤0, ∀x. (2.19)