v2007.11.26 - Convex Optimization

518 APPENDIX A. LINEAR ALGEBRAA.5 eigen decompositionWhen a square matrix X ∈ R m×m is diagonalizable, [251,5.6] then⎡ ⎤w T1 m∑X = SΛS −1 = [ s 1 · · · s m ] Λ⎣. ⎦ = λ i s i wi T (1365)where s i ∈ C m are linearly independent (right-)eigenvectors A.12 constitutingthe columns of S ∈ C m×m defined byw T mi=1XS = SΛ (1366)w T i ∈ C m are linearly independent left-eigenvectors of X constituting the rowsof S −1 defined by [150]S −1 X = ΛS −1 (1367)and where {λ i ∈ C} are eigenvalues (populating diagonal matrix Λ∈ C m×m )corresponding to both left and right eigenvectors; id est, λ(X) = λ(X T ).There is no connection between diagonalizability and invertibility of X .[251,5.2] 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(1368)A.5.0.0.1 Theorem. Real eigenvector. Eigenvectors of a real matrixcorresponding 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 it follows: eigenvalues, includingtheir multiplicity, for a given square matrix are unique; meaning, there is noother set of eigenvalues for that matrix. (Conversely, many different matricesmay share the same unique set of eigenvalues.)Uniqueness of eigenvectors, in contrast, disallows multiplicity of the samedirection.A.12 Eigenvectors must, of course, be nonzero. The prefix eigen is from the German; inthis context meaning, something akin to “characteristic”. [248, p.14]