v2009.01.01 - Convex Optimization

E.6. VECTORIZATION INTERPRETATION, 663
E.6.2.1.1 Example. λ as coefficients of nonorthogonal projection.
Any diagonalization (A.5)
X = SΛS −1 =
m∑
λ i s i wi T ∈ R m×m (1438)
i=1
may be expressed as a sum of one-dimensional nonorthogonal projections
of X , each on the range of a vectorized eigenmatrix P j ∆ = s j w T j ;
∑
X = m 〈(Se i e T jS −1 ) T , X〉 Se i e T jS −1
i,j=1
∑
= m ∑
〈(s j wj T ) T , X〉 s j wj T + m 〈(Se i e T jS −1 ) T , SΛS −1 〉 Se i e T jS −1
j=1
∑
= m 〈(s j wj T ) T , X〉 s j wj
T
j=1
i,j=1
j ≠ i
∆ ∑
= m ∑
〈Pj T , X〉 P j = m s j wj T Xs j wj
T
j=1
∑
= m λ j s j wj
T
j=1
j=1
∑
= m P j XP j
j=1
(1845)
This biorthogonal expansion of matrix X is a sum of nonorthogonal
projections because the term outside the projection coefficient 〈 〉 is not
identical to the inside-term. (E.6.4) The eigenvalues λ j are coefficients of
nonorthogonal projection of X , while the remaining M(M −1)/2 coefficients
(for i≠j) are zeroed by projection. When P j is rank-one as in (1845),
R(vec P j XP j ) = R(vecs j w T j ) = R(vecP j ) in R m2 (1846)
and
P j XP j − X ⊥ P T j in R m2 (1847)
Were matrix X symmetric, then its eigenmatrices would also be. So the
one-dimensional projections would become orthogonal. (E.6.4.1.1)