v2010.10.26 - Convex Optimization

618 APPENDIX A. LINEAR ALGEBRAA.5.0.1.1 Definition. Unique eigenvectors.When eigenvectors are unique, we mean: unique to within a real nonzeroscaling, and their directions are distinct.△If S is a matrix of eigenvectors of X as in (1547), for example, then −Sis certainly another matrix of eigenvectors decomposing X with the sameeigenvalues.For any square matrix, the eigenvector corresponding to a distincteigenvalue is unique; [328, p.220]distinct eigenvalues ⇒ eigenvectors unique (1552)Eigenvectors corresponding to a repeated eigenvalue are not unique for adiagonalizable matrix;repeated eigenvalue ⇒ eigenvectors not unique (1553)Proof follows from the observation: any linear combination of distincteigenvectors of diagonalizable X , corresponding to a particular eigenvalue,produces another eigenvector. For eigenvalue λ whose multiplicity A.12dim N(X −λI) exceeds 1, in other words, any choice of independentvectors from N(X −λI) (of the same multiplicity) constitutes eigenvectorscorresponding to λ .Caveat diagonalizability insures linear independence which impliesexistence of distinct eigenvectors. We may conclude, for diagonalizablematrices,distinct eigenvalues ⇔ eigenvectors unique (1554)A.5.0.2Invertible matrixWhen diagonalizable matrix X ∈ R m×m is nonsingular (no zero eigenvalues),then it has an inverse obtained simply by inverting eigenvalues in (1547):X −1 = SΛ −1 S −1 (1555)A.12 A matrix is diagonalizable iff algebraic multiplicity (number of occurrences of sameeigenvalue) equals geometric multiplicity dim N(X −λI) = m − rank(X −λI) [328, p.15](number of Jordan blocks w.r.t λ or number of corresponding l.i. eigenvectors).