v2009.01.01 - Convex Optimization

570 APPENDIX A. LINEAR ALGEBRA
A.7.2
0 entry
If a positive semidefinite matrix A = [A ij ] ∈ R n×n has a 0 entry A ii on its
main diagonal, then A ij + A ji = 0 ∀j . [235,1.3.1]
Any symmetric positive semidefinite matrix having a 0 entry on its main
diagonal must be 0 along the entire row and column to which that 0 entry
belongs. [134,4.2.8] [176,7.1, prob.2]
A.7.3 0 eigenvalues theorem
This theorem is simple, powerful, and widely applicable:
A.7.3.0.1 Theorem. Number of 0 eigenvalues.
For any matrix A∈ R m×n
rank(A) + dim N(A) = n (1469)
by conservation of dimension. [176,0.4.4]
For any square matrix A∈ R m×m , the number of 0 eigenvalues is at least
equal to dim N(A)
dim N(A) ≤ number of 0 eigenvalues ≤ m (1470)
while the eigenvectors corresponding to those 0 eigenvalues belong to N(A).
[287,5.1] A.16
For diagonalizable matrix A (A.5), the number of 0 eigenvalues is
precisely dim N(A) while the corresponding eigenvectors span N(A). The
real and imaginary parts of the eigenvectors remaining span R(A).
A.16 We take as given the well-known fact that the number of 0 eigenvalues cannot be less
than the dimension of the nullspace. We offer an example of the converse:
A =
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1 0 1 0
0 0 1 0
0 0 0 0
1 0 0 0
dim N(A) = 2, λ(A) = [0 0 0 1] T ; three eigenvectors in the nullspace but only two are
independent. The right-hand side of (1470) is tight for nonzero matrices; e.g., (B.1) dyad
uv T ∈ R m×m has m 0-eigenvalues when u ∈ v ⊥ .
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