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f(A)-E\vert$的形式,即求$f(A)$的特征值。 + +\textbf{例题:}设$A$为三阶矩阵,已知$-3E+A$不可逆,$\vert2E+A\vert=0$,$(E-A)x=0$有非零解,求$\vert A^*-E\vert$。 + +解:前三个条件都是为了指明$\vert-3E+A\vert=\vert3E-A\vert=0$,$\vert-2E-A\vert=0$,$\vert E-A\vert=0$,即得到$A$的三个特征值$\lambda_1=3$、$\lambda_2=-2$、$\lambda_3=1$。 + +求$\vert A^*-E\vert$即求$A^*$的特征值,然后再乘起来,即得到行列式的值。 + +又$A^{-1}=\dfrac{A^*}{\vert A\vert}$,所以$A^*=\dfrac{\vert A\vert}{A}$。 + +$\vert A\vert$等于特征值的乘积$-6$,对应的特征值即为$\mu_1=\dfrac{-6}{3}=-2$,$\mu_2=\dfrac{-6}{-2}=3$,$\mu_3=\dfrac{-6}{1}=-6$。 + +对应$A^*-E$的特征值为$-3$,$2$,$-6$,所以最后的行列式值为$42$。 + +\subsubsection{矩阵函数} + +题目要求$\vert f(A)\vert$的形式,即求$f(A)$的特征值,然后求其乘积就是矩阵方程的行列式值。 + \section{相似理论} $P^{-1}AP=\Lambda$,$P$为特征向量组,$\Lambda$为特征值矩阵。 diff --git a/linear-algebra/exercise/6-quadratic-form/quadratic-form.pdf b/linear-algebra/exercise/6-quadratic-form/quadratic-form.pdf index 35ca58b41bd71c1c6510007ec6eb9189c0862f45..0051d3240b9a77ee52d0a69c32ab4dab42c9e00a 100644 GIT binary patch delta 73170 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