v2010.10.26 - Convex Optimization

A.3. PROPER STATEMENTS 603For A diagonalizable (A.5), A = SΛS −1 , (confer [331, p.255])rankA = rankδ(λ(A)) = rank Λ (1460)meaning, rank is equal to the number of nonzero eigenvalues in vectorby the 0 eigenvalues theorem (A.7.3.0.1).(Ky Fan) For A,B∈ S n [55,1.2] (confer (1741))λ(A) δ(Λ) (1461)tr(AB) ≤ λ(A) T λ(B) (1727)with equality (Theobald) when A and B are simultaneouslydiagonalizable [202] with the same ordering of eigenvalues.For A∈ R m×n and B ∈ R n×mtr(AB) = tr(BA) (1462)and η eigenvalues of the product and commuted product are identical,including their multiplicity; [202,1.3.20] id est,λ(AB) 1:η = λ(BA) 1:η , ηmin{m , n} (1463)Any eigenvalues remaining are zero. By the 0 eigenvalues theorem(A.7.3.0.1),rank(AB) = rank(BA), AB and BA diagonalizable (1464)For any compatible matrices A,B [202,0.4]min{rankA, rankB} ≥ rank(AB) (1465)For A,B ∈ S n +rankA + rankB ≥ rank(A + B) ≥ min{rankA, rankB} ≥ rank(AB)(1466)For linearly independent matrices A,B ∈ S n + (2.1.2, R(A) ∩ R(B)=0,R(A T ) ∩ R(B T )=0,B.1.1),rankA + rankB = rank(A + B) > min{rankA, rankB} ≥ rank(AB)(1467)