Return to Revisions

If $e$ is a small number, then $od(H)\approx od(A)+e(od)'_A(B)$. $od(A)=1-\dfrac{u(A)}{v(A)}$ where $u(A)=\det(A^*A),v(A)=\Pi_i||Ae_i||^2$ and $(e_i)_i$ is the canonical basis. $(od)'_A=-\dfrac{1}{v(A)}u'_A+\dfrac{u(A)}{v^2(A)}v'_A$. $u'_A(K)=trace((A^*K+K^*A)adjoint(A^*A))$. $v'_A(K)=\sum_i((e_i^*K^*Ae_i+e_i^*A^*Ke_i)\Pi_{j\not= i}||Ae_j||^2)$