v2006.03.09 - Convex Optimization

A.5. EIGEN DECOMPOSITION 449A.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 (1144), for example, then −Sis certainly another matrix of eigenvectors decomposing X with the sameeigenvalues.For any square matrix, [209, p.220]distinct eigenvalues ⇒ eigenvectors unique (1148)Eigenvectors corresponding to a repeated eigenvalue are not unique for adiagonalizable matrix;repeated eigenvalue ⇒ eigenvectors not unique (1149)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.13dim 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 (1150)A.5.1EigenmatrixThe (right-)eigenvectors {s i } are naturally orthogonal to the left-eigenvectors{w i } except, for i = 1... m , w T i s i = 1 ; called a biorthogonality condition[238,2.2.4] [125] because neither set of left or right eigenvectors is necessarilyan orthogonal set. Consequently, each dyad from a diagonalization is anindependent (B.1.1) nonorthogonal projector becauseA.13 For a diagonalizable matrix, algebraic multiplicity is the same as geometric multiplicity.[209, p.15]