v2009.01.01 - Convex Optimization

52 CHAPTER 2. CONVEX GEOMETRY
where all entries off the main diagonal have been scaled. Now for Z ∈ S M
〈Y , Z〉 ∆ = tr(Y T Z) = vec(Y ) T vec Z = svec(Y ) T svec Z (50)
Then because the metrics become equivalent, for X ∈ S M
‖ svec X − svec Y ‖ 2 = ‖X − Y ‖ F (51)
and because symmetric vectorization (49) is a linear bijective mapping, then
svec is an isometric isomorphism of the symmetric matrix subspace. In other
words, S M is isometrically isomorphic with R M(M+1)/2 in the Euclidean sense
under transformation svec .
The set of all symmetric matrices S M forms a proper subspace in R M×M ,
so for it there exists a standard orthonormal basis in isometrically isomorphic
R M(M+1)/2 ⎧
⎨
{E ij ∈ S M } =
⎩
e i e T i ,
1 √
2
(e i e T j + e j e T i
i = j = 1... M
)
, 1 ≤ i < j ≤ M
⎫
⎬
⎭
(52)
where M(M + 1)/2 standard basis matrices E ij are formed from the
standard basis vectors
[{ ]
1, i = j
e i =
0, i ≠ j , j = 1... M ∈ R M (53)
Thus we have a basic orthogonal expansion for Y ∈ S M
Y =
M∑ j∑
〈E ij ,Y 〉 E ij (54)
j=1 i=1
whose coefficients
〈E ij ,Y 〉 =
{
Yii , i = 1... M
Y ij
√
2 , 1 ≤ i < j ≤ M
(55)
correspond to entries of the symmetric vectorization (49).