谱定理

有限维的情况

埃尔米特矩阵

${\displaystyle \langle Ax,\,y\rangle =\langle x,\,Ay\rangle }$

证明

${\displaystyle V_{\lambda }=\{\,v\in V:Av=\lambda v\,\}}$

${\displaystyle A=\lambda _{1}P_{\lambda _{1}}+\cdots +\lambda _{m}P_{\lambda _{m}}.}$

正规矩阵

${\displaystyle A=U\Lambda U^{*}\;}$

有界自伴算子的谱定理

${\displaystyle [A\varphi ](t)=t\varphi (t).\;}$

${\displaystyle U^{*}TU=A\;}$

${\displaystyle [T\varphi ](x)=f(x)\varphi (x).\;}$

参考

• Sheldon Axler, Linear Algebra Done Right, Springer Verlag, 1997
• Paul Halmos, "What Does the Spectral Theorem Say?", American Mathematical Monthly, volume 70, number 3 (1963), pages 241–247