v2009.01.01 - Convex Optimization

A.5. EIGEN DECOMPOSITION 561
When B is full-rank and fat, A = 0, and C ≽ 0, then
detG ≠ 0 ⇔ C + B T B ≻ 0 (1435)
When B is a row-vector, then for A ≠ 0 and all C of dimension
compatible with G
while for all A∈ R
detG = A det(C − 1 A BT B) (1436)
detG = det(C)A − BC T cofB T (1437)
where C cof is the matrix of cofactors corresponding to C .
A.5 eigen decomposition
When a square matrix X ∈ R m×m is diagonalizable, [287,5.6] then
⎡ ⎤
w T1 m∑
X = SΛS −1 = [s 1 · · · s m ] Λ⎣
. ⎦ = λ i s i wi T (1438)
where {s i ∈ N(X − λ i I)⊆ C m } are l.i. (right-)eigenvectors A.12 constituting
the columns of S ∈ C m×m defined by
w T m
i=1
XS = SΛ (1439)
{w i ∈ N(X T − λ i I)⊆ C m } are linearly independent left-eigenvectors of X
(eigenvectors of X T ) constituting the rows of S −1 defined by [176]
S −1 X = ΛS −1 (1440)
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 .
[287,5.2] Diagonalizability is guaranteed by a full set of linearly independent
eigenvectors, whereas invertibility is guaranteed by all nonzero eigenvalues.
distinct eigenvalues ⇒ l.i. eigenvectors ⇔ diagonalizable
not diagonalizable ⇒ repeated eigenvalue
(1441)
A.12 Eigenvectors must, of course, be nonzero. The prefix eigen is from the German; in
this context meaning, something akin to “characteristic”. [285, p.14]