v2010.10.26 - Convex Optimization

E.6. VECTORIZATION INTERPRETATION, 727eigenmatrices q i qiT of C 1 by applying a similarity transformation; [331,5.6]{ }qi q{QE ij Q T i T , i = j = 1... M} = ( ) (1982)√1qi 2qj T + q j qiT , 1 ≤ i < j ≤ Mwhich remains an orthonormal basis for S M . Then remarkably∑C 1 = M 〈QE ij Q T , C 1 〉 QE ij Q Ti,j=1j ≥ i∑= M ∑〈q i qi T , C 1 〉 q i qi T + M 〈QE ij Q T , QΛQ T 〉 QE ij Q Ti=1∑= M 〈q i qi T , C 1 〉 q i qiTi=1∑ Mi=1i,j=1j > i∑〈P i , C 1 〉 P i = M q i qi T C 1 q i qiT∑= M λ i q i qiTi=1i=1∑= M P i C 1 P ii=1(1983)this orthogonal expansion becomes the diagonalization; still a sum ofone-dimensional orthogonal projections. The eigenvaluesλ i = 〈q i q T i , C 1 〉 (1984)are clearly coefficients of projection of C 1 on the range of each vectorizedeigenmatrix. (conferE.6.2.1.1) The remaining M(M −1)/2 coefficients(i≠j) are zeroed by projection. When P i is rank-one symmetric as in (1983),R(svecP i C 1 P i ) = R(svec q i q T i ) = R(svec P i ) in R M(M+1)/2 (1985)andP i C 1 P i − C 1 ⊥ P i in R M(M+1)/2 (1986)E.6.4.2Positive semidefiniteness test as orthogonal projectionFor any given X ∈ R m×m the familiar quadratic construct y T Xy ≥ 0,over broad domain, is a fundamental test for positive semidefiniteness.