v2010.10.26 - Convex Optimization

2.5. SUBSPACE REPRESENTATIONS 91are partially defining. But to aid visualization of involved geometry, itsometimes helps to vectorize matrices. For any square matrix A , s∈ N(A) ,and w ∈ N(A T )〈A, ss T 〉 = 0, 〈A, ww T 〉 = 0 (161)because s T As = w T Aw = 0. This innocuous observation becomes a sharpinstrument for visualization of diagonalizable matrices (A.5.1): for rank-ρmatrix A∈ R M×MA = SΛS −1 = [ s 1 · · · s M ] Λ⎣⎡w T1.w T M⎤⎦ =M∑λ i s i wi T (1547)where nullspace eigenvectors are real by theorem A.5.0.0.1 and where (B.1.1)( )M∑R{s i ∈ R M |λ i =0} = R s i s T i = N(A)i=ρ+1( )(162)M∑R{w i ∈ R M |λ i =0} = R w i wiT = N(A T )i=ρ+1Define an unconventional basis among column vectors of each summation:i=1basis N(A) ⊆basis N(A T ) ⊆M ∑i=ρ+1∑ Mi=ρ+1s i s T i⊆ N(A)w i w T i ⊆ N(A T )(163)We shall regard a vectorized subspace as vectorization of any M ×M matrixwhose columns comprise an overcomplete basis for that subspace; e.g.,E.3.1∑vec basis N(A) = vec M s i s T ii=ρ+1∑vec basis N(A T ) = vec M w i wiTi=ρ+1(164)By this reckoning, vec basis R(A)= vec A but is not unique. Now, because〈 〉 〈 〉M∑M∑A , s i s T i = 0, A , w i wiT = 0 (165)i=ρ+1i=ρ+1