v2007.09.13 - Convex Optimization

606 APPENDIX E. PROJECTIONq i q T iof C 1 by applying a similarity transformation; [247,5.6]{ }qi q{QE ij Q T i T , i = j = 1... M} = ( )√1qi 2qj T + q j qiT , 1 ≤ i < j ≤ M(1745)which 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=1i,j=1j > i∆ ∑= M ∑〈P i , C 1 〉 P i = M q i qi T C 1 q i qiTi=1∑= M λ i q i qiTi=1i=1∑= M P i C 1 P ii=1(1746)this orthogonal expansion becomes the diagonalization; still a sum ofone-dimensional orthogonal projections. The eigenvaluesλ i = 〈q i q T i , C 1 〉 (1747)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 (1746),R(svecP i C 1 P i ) = R(svecq i q T i ) = R(svecP i ) in R M(M+1)/2 (1748)andP i C 1 P i − C 1 ⊥ P i in R M(M+1)/2 (1749)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.