v2009.01.01 - Convex Optimization

656 APPENDIX E. PROJECTION
and define the rows a ∗T
i of its pseudoinverse A † as in (1803). Then
orthogonal projection of vector x∈ R n on R(A) is a sum of one-dimensional
nonorthogonal projections
Px = AA † x =
n∑
i=1
a i a ∗T
i x (1813)
where each nonsymmetric dyad a i a ∗T
i is a nonorthogonal projector projecting
on R(a i ) , (E.6.1) idempotent because of biorthogonality condition A † A = I .
The projection Px is regarded as the best approximation to x from the
set R(A) , as it was in Example E.5.0.0.1.
E.5.0.0.3 Example. Biorthogonal expansion as nonorthogonal projection.
Biorthogonal expansion can be viewed as a sum of components, each a
nonorthogonal projection on the range of an extreme direction of a pointed
polyhedral cone K ; e.g., Figure 139.
Suppose matrix A∈ R m×n holds a nonorthogonal basis for R(A) in
its columns as in (1802), and the rows of pseudoinverse A † are defined
as in (1803). Assuming the most general biorthogonality condition
(A † + BZ T )A = I with BZ T defined as for (1762), then biorthogonal
expansion of vector x is a sum of one-dimensional nonorthogonal projections;
for x∈ R(A)
x = A(A † + BZ T )x = AA † x =
n∑
i=1
a i a ∗T
i x (1814)
where each dyad a i a ∗T
i is a nonorthogonal projector projecting on R(a i ).
(E.6.1) The extreme directions of K=cone(A) are {a 1 ,..., a n } the linearly
independent columns of A , while the extreme directions {a ∗ 1 ,..., a ∗ n}
of relative dual cone K ∗ ∩aff K=cone(A †T ) (2.13.9.4) correspond to the
linearly independent (B.1.1.1) rows of A † . Directions of nonorthogonal
projection are determined by the pseudoinverse; id est, direction of
projection a i a ∗T
i x−x on R(a i ) is orthogonal to a ∗ i . E.9
E.9 This remains true in high dimension although only a little more difficult to visualize
in R 3 ; confer , Figure 56.