v2007.11.26 - Convex Optimization

162 CHAPTER 2. CONVEX GEOMETRYcoordinates for biorthogonal expansion are its eigenvalues λ i (contained indiagonal matrix Λ) when expanded in S ;⎡X = SS −1 X = [s 1 · · · s M ] ⎣⎤w1 T X. ⎦ =wM T XM∑λ i s i wi T (340)Coordinate value depend upon the geometric relationship of X to its linearlyindependent eigenmatrices s i w T i . (A.5.1,B.1.1)Eigenmatrices s i w T i are linearly independent dyads constituted by rightand left eigenvectors of diagonalizable X and are generators of somepointed polyhedral cone K in a subspace of R M×M .When S is real and X belongs to that polyhedral cone K , for example,then coordinates of expansion (the eigenvalues λ i ) must be nonnegative.When X = QΛQ T is symmetric, coordinates for biorthogonal expansionare its eigenvalues when expanded in Q ; id est, for X ∈ S Mi=1X = QQ T X =M∑q i qi T X =i=1M∑λ i q i qi T ∈ S M (341)i=1becomes an orthogonal expansion with orthonormality condition Q T Q=Iwhere λ i is the i th eigenvalue of X , q i is the corresponding i th eigenvectorarranged columnar in orthogonal matrixQ = [q 1 q 2 · · · q M ] ∈ R M×M (342)and where eigenmatrix q i qiT is an extreme direction of some pointedpolyhedral cone K ⊂ S M and an extreme direction of the positive semidefinitecone S M + .Orthogonal expansion is a special case of biorthogonal expansion ofX ∈ aff K occurring when polyhedral cone K is any rotation about theorigin of an orthant belonging to a subspace.Similarly, when X = QΛQ T belongs to the positive semidefinite cone inthe subspace of symmetric matrices, coordinates for orthogonal expansion