v2009.01.01 - Convex Optimization

E.3. SYMMETRIC IDEMPOTENT MATRICES 653
E.3.5.1
orthogonal projector, orthonormal decomposition
Consider orthogonal expansion of x∈ R(U) :
n∑
x = UU T x = u i u T i x (1798)
a sum of one-dimensional orthogonal projections (E.6.3), where
i=1
U ∆ = [u 1 · · · u n ] and U T U = I (1799)
and where the subspace projector has two expressions, (1797)
AA † ∆ = UU T (1800)
where A ∈ R m×n has rank n . The direction of projection of x on u j for
some j ∈{1... n} , for example, is orthogonal to u j but parallel to a vector
in the span of all the remaining vectors constituting the columns of U ;
u T j(u j u T j x − x) = 0
u j u T j x − x = u j u T j x − UU T x ∈ R({u i |i=1... n, i≠j})
(1801)
E.3.5.2
orthogonal projector, biorthogonal decomposition
We get a similar result for the biorthogonal expansion of x∈ R(A). Define
A ∆ = [a 1 a 2 · · · a n ] ∈ R m×n (1802)
and the rows of the pseudoinverse E.8
⎡ ⎤
a ∗T
1
A † =
∆ a ∗T
⎢
⎣
2 ⎥
. ⎦ ∈ Rn×m (1803)
a ∗T
n
under biorthogonality condition A † A=I . In biorthogonal expansion
(2.13.8)
n∑
x = AA † x = a i a ∗T
i x (1804)
E.8 Notation * in this context connotes extreme direction of a dual cone; e.g., (361) or
Example E.5.0.0.3.
i=1