v2009.01.01 - Convex Optimization

A.5. EIGEN DECOMPOSITION 563
dim N(X −λI) exceeds 1, in other words, any choice of independent
vectors from N(X −λI) (of the same multiplicity) constitutes eigenvectors
corresponding to λ .
Caveat diagonalizability insures linear independence which implies
existence of distinct eigenvectors. We may conclude, for diagonalizable
matrices,
distinct eigenvalues ⇔ eigenvectors unique (1444)
A.5.1
eigenmatrix
The (right-)eigenvectors {s i } (1438) are naturally orthogonal w T i s j =0
to left-eigenvectors {w i } except, for i=1... m , w T i s i =1 ; called a
biorthogonality condition [318,2.2.4] [176] because neither set of left or right
eigenvectors is necessarily an orthogonal set. Consequently, each dyad from a
diagonalization is an independent (B.1.1) nonorthogonal projector because
s i w T i s i w T i = s i w T i (1445)
(whereas the dyads of singular value decomposition are not inherently
projectors (confer (1450))).
The dyads of eigen decomposition can be termed eigenmatrices because
Sum of the eigenmatrices is the identity;
X s i w T i = λ i s i w T i (1446)
m∑
s i wi T = I (1447)
i=1
A.5.2
Symmetric matrix diagonalization
The set of normal matrices is, precisely, that set of all real matrices having
a complete orthonormal set of eigenvectors; [344,8.1] [289, prob.10.2.31]
id est, any matrix X for which XX T = X T X ; [134,7.1.3] [285, p.3]
e.g., orthogonal and circulant matrices [142]. All normal matrices are
diagonalizable. A symmetric matrix is a special normal matrix whose