v2009.01.01 - Convex Optimization

562 APPENDIX A. LINEAR ALGEBRA
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.5.0.1
Uniqueness
From the fundamental theorem of algebra it follows: eigenvalues, including
their multiplicity, for a given square matrix are unique; meaning, there is
no other set of eigenvalues for that matrix. (Conversely, many different
matrices may share 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 same
direction:
A.5.0.1.1 Definition. Unique eigenvectors.
When eigenvectors are unique, we mean: unique to within a real nonzero
scaling, and their directions are distinct.
△
If S is a matrix of eigenvectors of X as in (1438), for example, then −S
is certainly another matrix of eigenvectors decomposing X with the same
eigenvalues.
For any square matrix, the eigenvector corresponding to a distinct
eigenvalue is unique; [285, p.220]
distinct eigenvalues ⇒ eigenvectors unique (1442)
Eigenvectors corresponding to a repeated eigenvalue are not unique for a
diagonalizable matrix;
repeated eigenvalue ⇒ eigenvectors not unique (1443)
Proof follows from the observation: any linear combination of distinct
eigenvectors of diagonalizable X , corresponding to a particular eigenvalue,
produces another eigenvector. For eigenvalue λ whose multiplicity A.13
A.13 A matrix is diagonalizable iff algebraic multiplicity (number of occurrences of same
eigenvalue) equals geometric multiplicity dim N(X −λI) = m − rank(X −λI) [285, p.15]
(number of Jordan blocks w.r.t λ or number of corresponding l.i. eigenvectors).