v2010.10.26 - Convex Optimization

5.11. EDM INDEFINITENESS 4695.11.1 EDM eigenvalues, congruence transformationFor any symmetric −D , we can characterize its eigenvalues by congruencetransformation: [331,6.3][ ]VT−W T NDW = − D [V N1 T1 ] = −[VTN DV N V T N D11 T DV N 1 T D1]∈ S N (1100)BecauseW [V N 1 ] ∈ R N×N (1101)is full-rank, then (1516)inertia(−D) = inertia ( −W T DW ) (1102)the congruence (1100) has the same number of positive, zero, and negativeeigenvalues as −D . Further, if we denote by {γ i , i=1... N −1} theeigenvalues of −VN TDV N and denote eigenvalues of the congruence −W T DWby {ζ i , i=1... N} and if we arrange each respective set of eigenvalues innonincreasing order, then by theory of interlacing eigenvalues for borderedsymmetric matrices [202,4.3] [331,6.4] [328,IV.4.1]ζ N ≤ γ N−1 ≤ ζ N−1 ≤ γ N−2 ≤ · · · ≤ γ 2 ≤ ζ 2 ≤ γ 1 ≤ ζ 1 (1103)When D ∈ EDM N , then γ i ≥ 0 ∀i (1450) because −V T N DV N ≽0 as weknow. That means the congruence must have N −1 nonnegative eigenvalues;ζ i ≥ 0, i=1... N −1. The remaining eigenvalue ζ N cannot be nonnegativebecause then −D would be positive semidefinite, an impossibility; so ζ N < 0.By congruence, nontrivial −D must therefore have exactly one negativeeigenvalue; 5.50 [113,2.4.5]5.50 All entries of the corresponding eigenvector must have the same sign, with respectto each other, [89, p.116] because that eigenvector is the Perron vector corresponding tospectral radius; [202,8.3.1] the predominant characteristic of square nonnegative matrices.Unlike positive semidefinite matrices, nonnegative matrices are guaranteed only to haveat least one nonnegative eigenvalue.