性质3设n阶方阵A=(a)特征值为1,2，，n,则 (①)+2+…+m=a11+22+…+am 41+2+…+am称为矩阵的迹，fr(A) (2)12…2m=A (3)AH=(a)n特征值为万，万，…，元 f2)=2E-A=(2-2-2)…(2-n) =20-(2+2+…+2n)2-1+…+(-1)”22…2n 20-(a1+2++am)2-1++-IA This docur at is produced by trial version of Print2Flash.Visit www.print2flash.com for m
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性质4设2，乙是A的2个不同特征值， ,P是对应于2，乙，的特征向量，则C,P,线性无关。 证设有一组数k,k2,kP+kB=0(①) 两边左乘A,A(kB+kD)=0=kAP+kB ①)乘2：kP+kB=0→kD(2-2)=0 又2≠元→k=0→k=0 结论设21，乙2，…，1n是方阵的m个特征值，P,P2: …,pn依次是与之对应的特征向量.如果1,2，…，m 各不相等，则P1,P2,…,Pm线性无关 ed by trial v of Print2Flash
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注意 1.属于不同特征值的特征向量是线性无关的. 2.属于同一特征值的特征向量的非零线性 组合仍是属于这个特征值的特征向量. 3. 矩阵的特征向量总是相对于矩阵的特征值而 言的，一个特征值具有的特征向量不唯一； 一个特征向量不能属于不同的特征值. 4. 特征值的几何重数不大于它的代数重数。 uced by trial ver f Print2Flash.Visit www.print2flash.con
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5.设2,2，…，九是A的r个互不同的特征 值，九，的几何重数为qi,C1,C2,…,Cg是对 应于入，的9，个线性无关的特征向量，则的所有这 些特征向量 41,0%12,C1g氵 0C21,022,,02g2 0r1,Cp2,,0g 仍然是线性无关的。 his do Is pro ed by trial v
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n阶矩阵A是奇异矩阵的充分必要条件 是A有一个特征值为零。 设A,B∈C",则tr(AB)=tr（BD· his doc ced by trial v Visit www.print2flash.com for mo
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