v2009.01.01 - Convex Optimization

5.11. EDM INDEFINITENESS 417
5.11.1 EDM eigenvalues, congruence transformation
For any symmetric −D , we can characterize its eigenvalues by congruence
transformation: [287,6.3]
[ ]
V
T
−W T N
DW = − D [ V N
1 T
1 ] = −
[
V
T
N DV N V T N D1
1 T DV N 1 T D1
]
∈ S N (1005)
Because
W ∆ = [V N 1 ] ∈ R N×N (1006)
is full-rank, then (1415)
inertia(−D) = inertia ( −W T DW ) (1007)
the congruence (1005) has the same number of positive, zero, and negative
eigenvalues as −D . Further, if we denote by {γ i , i=1... N −1} the
eigenvalues of −VN TDV N and denote eigenvalues of the congruence −W T DW
by {ζ i , i=1... N} and if we arrange each respective set of eigenvalues in
nonincreasing order, then by theory of interlacing eigenvalues for bordered
symmetric matrices [176,4.3] [287,6.4] [285,IV.4.1]
ζ N ≤ γ N−1 ≤ ζ N−1 ≤ γ N−2 ≤ · · · ≤ γ 2 ≤ ζ 2 ≤ γ 1 ≤ ζ 1 (1008)
When D ∈ EDM N , then γ i ≥ 0 ∀i (1353) because −V T N DV N ≽0 as we
know. That means the congruence must have N −1 nonnegative eigenvalues;
ζ i ≥ 0, i=1... N −1. The remaining eigenvalue ζ N cannot be nonnegative
because then −D would be positive semidefinite, an impossibility; so ζ N < 0.
By congruence, nontrivial −D must therefore have exactly one negative
eigenvalue; 5.46 [96,2.4.5]
5.46 All the entries of the corresponding eigenvector must have the same sign with respect
to each other [78, p.116] because that eigenvector is the Perron vector corresponding to
the spectral radius; [176,8.2.6] the predominant characteristic of negative [sic] matrices.