5. 设 $A,B\in M_n$, 则 $$\bex s_j(AB)\leq \sen{A}_\infty s_j(B),\quad s_j(AB)\leq \sen{B}_\infty s_j(A),\quad j=1,\cdots,n. \eex$$

证明: 由定理 4.3 (Ky Fan), $$\bex s_j(AB)\leq s_1(A)s_j(B)=\sen{A}_\infty s_j(B), \eex$$ $$\bex s_j(BA)=s_j(BA)\leq s_1(B)s_j(A)=\sen{B}_\infty s_j(A). \eex$$