对任两酉阵 $U,V$, 有 $$ex sen{A}_F=sen{UAV}_F. eex$$
事实上, $$eex ea sen{UAV}_F^2&= r(V^*A^*U^*cdot UAV)\ &= r (V^*A^*AV)\ &= r(AVV^*A^*)quadsex{ r(AB)= r(BA)}\ &= r(AA^*)\ &= r(A^*A)\ &=sen{A}_F^2. eea eeex$$
对任两酉阵 $U,V$, 有 $$ex sen{A}_F=sen{UAV}_F. eex$$
事实上, $$eex ea sen{UAV}_F^2&= r(V^*A^*U^*cdot UAV)\ &= r (V^*A^*AV)\ &= r(AVV^*A^*)quadsex{ r(AB)= r(BA)}\ &= r(AA^*)\ &= r(A^*A)\ &=sen{A}_F^2. eea eeex$$