Prove that for any matrices $A,B$ we have $$ex |per (AB)|^2leq per (AA^*)cdot per (B^*B). eex$$ (The corresponding relation for determinants is an easy equality.)
Solution. Let $$ex A=sex{a{cc} al_1\ vdots\ al_n ea},quad B=sex{eta_1,cdots,eta_n}. eex$$ Then $$ex AB=sex{sef{al_i,eta_j}}. eex$$ By Exercise I.5.7, $$eex ea |per (AB)|^2 &=sev{per (sef{al_i,eta_j})}^2\ &leq per (sef{al_i,al_j})cdot per (sef{eta_i,eta_j})\ &=per(AA^*)cdot per(B^*B). eea eeex$$