v2010.10.26 - Convex Optimization

620 APPENDIX A. LINEAR ALGEBRAwhere 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 . By 0 eigenvalues theoremA.7.3.0.1,R{s i |λ i ≠0} = R(A) = R(A T )R{s i |λ i =0} = N(A T (1560)) = N(A)A.5.1.1Diagonal matrix diagonalizationBecause arrangement of eigenvectors and their corresponding eigenvalues isarbitrary, we almost always arrange eigenvalues in nonincreasing order asis the convention for singular value decomposition. Then to diagonalize asymmetric matrix that is already a diagonal matrix, orthogonal matrix Sbecomes a permutation matrix.A.5.1.2Invertible symmetric matrixWhen symmetric matrix X ∈ S m is nonsingular (invertible), then its inverse(obtained by inverting eigenvalues in (1559)) is also symmetric:A.5.1.3X −1 = SΛ −1 S T ∈ S m (1561)Positive semidefinite matrix square rootWhen X ∈ S m + , its unique positive semidefinite matrix square root is defined√X S√ΛS T ∈ S m + (1562)where the square root of nonnegative diagonal matrix √ Λ is taken entrywiseand positive. Then X = √ X √ X .A.6 Singular value decomposition, SVDA.6.1Compact SVD[159,2.5.4] For any A∈ R m×n⎡q T1A = UΣQ T = [ u 1 · · · u η ] Σ⎣.⎤∑⎦ = η σ i u i qiTqηT i=1(1563)U ∈ R m×η , Σ ∈ R η×η , Q ∈ R n×η