v2009.01.01 - Convex Optimization

E.1. IDEMPOTENT MATRICES 645
order, and where s i ,w i ∈ R m are the right- and left-eigenvectors of P ,
respectively, which are independent and real. E.5 Therefore
U ∆ = S(:,1:k) = [ s 1 · · · s k
]
∈ R
m×k
(1769)
is the full-rank matrix S ∈ R m×m having m − k columns truncated
(corresponding to 0 eigenvalues), while
⎡
Q T = ∆ S −1 (1:k, :) = ⎣
w T1
.
w T k
⎤
⎦ ∈ R k×m (1770)
is matrix S −1 having the corresponding m − k rows truncated. By the
0 eigenvalues theorem (A.7.3.0.1), R(U)= R(P ) , R(Q)= R(P T ) , and
R(P ) = span {s i | φ i = 1 ∀i}
N(P ) = span {s i | φ i = 0 ∀i}
R(P T ) = span {w i | φ i = 1 ∀i}
N(P T ) = span {w i | φ i = 0 ∀i}
(1771)
Thus biorthogonality Q T U =I is a necessary condition for idempotence, and
so the collection of nonorthogonal projectors projecting on R(U) is the affine
subset P k =UQ T k where Q k = {Q | Q T U = I , Q∈ R m×k }.
(⇒) Biorthogonality is a sufficient condition for idempotence;
P 2 =
k∑
s i wi
T
i=1
k∑
s j wj T = P (1772)
j=1
id est, if the cross-products are annihilated, then P 2 =P .
Nonorthogonal projection of
biorthogonal expansion,
x on R(P ) has expression like a
Px = UQ T x =
k∑
wi T xs i (1773)
i=1
E.5 Eigenvectors of a real matrix corresponding to real eigenvalues must be real.
(A.5.0.0.1)