v2010.10.26 - Convex Optimization

A.5. EIGENVALUE DECOMPOSITION 617where {s i ∈ N(X − λ i I)⊆ C m } are l.i. (right-)eigenvectors constituting thecolumns of S ∈ C m×m defined byXS = SΛ rather Xs i λ i s i , i = 1... m (1548){w i ∈ N(X T − λ i I)⊆ C m } are linearly independent left-eigenvectors of X(eigenvectors of X T ) constituting the rows of S −1 defined by [202]S −1 X = ΛS −1 rather w T i X λ i w T i , i = 1... m (1549)and where {λ i ∈ C} are eigenvalues (1461)δ(λ(X)) = Λ ∈ C m×m (1550)corresponding to both left and right eigenvectors; id est, λ(X) = λ(X T ).There is no connection between diagonalizability and invertibility of X .[331,5.2] Diagonalizability is guaranteed by a full set of linearly independenteigenvectors, whereas invertibility is guaranteed by all nonzero eigenvalues.distinct eigenvalues ⇒ l.i. eigenvectors ⇔ diagonalizablenot diagonalizable ⇒ repeated eigenvalue(1551)A.5.0.0.1 Theorem. Real eigenvector.Eigenvectors of a real matrix corresponding to real eigenvalues must be real.⋄Proof. Ax = λx . Given λ=λ ∗ , x H Ax = λx H x = λ‖x‖ 2 = x T Ax ∗x = x ∗ , where x H =x ∗T . The converse is equally simple. ⇒A.5.0.1UniquenessFrom the fundamental theorem of algebra, [219] which guarantees existenceof zeros for a given polynomial, it follows: eigenvalues, including theirmultiplicity, for a given square matrix are unique; meaning, there is no otherset of eigenvalues for that matrix. (Conversely, many different matrices mayshare the same unique set of eigenvalues; e.g., for any X , λ(X) = λ(X T ).)Uniqueness of eigenvectors, in contrast, disallows multiplicity of the samedirection: