1. (Fan-Hoffman). 设 $A\in M_n$, 记 $\Re A=(A+A^*)/2$. 则 $$\bex \lm_j(\Re A)\leq s_j(A),\quad j=1,\cdots,n. \eex$$

 

 

 

证明: 对适合 $\sen{x}=1$ 的 $x\in\bbC^n$, $$\beex \bea x^*(\Re A)x&= x^*\frac{A+A^*}{2}x\\ &=\frac{1}{2}(x^*Ax+x^*A^*x)\\ &=\Re (x^*Ax)\quad\sex{z\in\bbC\ra z^*=\bar z}\\ &\leq |x^*Ax|\\ &=|\sef{Ax,x}|\\ &\leq \sen{Ax}. \eea \eeex$$ 于是由 Courant-Fischer 极小极大刻画, $$\beex \bea \lm_j(\Re A)&=\max_{S\subset \bbC^n\atop \dim S=j} \min_{x\in S\atop \sen{x}=1} x^*(\Re A)x\\ &\leq \max_{S\subset \bbC^n\atop \dim S=j} \min_{x\in S\atop \sen{x}=1} \sen{Ax}\\ &=s_j(A). \eea \eeex$$