v2007.09.17 - Convex Optimization

506 APPENDIX A. LINEAR ALGEBRAs i w T i s i w T i = s i w T i (1346)(whereas the dyads of singular value decomposition are not inherentlyprojectors (confer (1350))).The dyads of eigen decomposition can be termed eigenmatrices becauseA.5.2X s i w T i = λ i s i w T i (1347)Symmetric matrix diagonalizationThe set of normal matrices is, precisely, that set of all real matrices havinga complete orthonormal set of eigenvectors; [301,8.1] [251, prob.10.2.31]id est, any matrix X for which XX T = X T X ; [110,7.1.3] [246, p.3]e.g., orthogonal and circulant matrices [118]. All normal matrices arediagonalizable. A symmetric matrix is a special normal matrix whoseeigenvalues must be real and whose eigenvectors can be chosen to make areal orthonormal set; [251,6.4] [249, p.315] id est, for X ∈ S ms T1X = SΛS T = [ s 1 · · · s m ] Λ⎣.⎡s T m⎤⎦ =m∑λ i s i s T i (1348)where δ 2 (Λ) = Λ∈ S m (A.1) and S −1 = S T ∈ R m×m (orthogonal matrix,B.5) because of symmetry: SΛS −1 = S −T ΛS T .Because the arrangement of eigenvectors and their correspondingeigenvalues is arbitrary, we almost always arrange eigenvalues innonincreasing order as is the convention for singular value decomposition.Then to diagonalize a symmetric matrix that is already a diagonal matrix,orthogonal matrix S becomes a permutation matrix.i=1A.5.2.1Positive semidefinite matrix square rootWhen X ∈ S m + , its unique positive semidefinite matrix square root is defined√X ∆ = S √ ΛS T ∈ S m + (1349)where the square root of nonnegative diagonal matrix √ Λ is taken entrywiseand positive. Then X = √ X √ X .