CBMS-2015-08

题目来源：Question 8 日期：2024-06-15 题目主题：数学-线性代数-特征值与特征向量

具体题目

Let $Z^{\ast }=\left\{\left(\begin{array}{c}{z}_1\\ z_2\end{array}\right):\text{complex number},|z_1{|}^2+|z_2{|}^2\ne 0\right\}$ be the set of non-zero complex two-dimensional vectors. Let $M=\left(\begin{array}{cc}a& b\\ b& d\end{array}\right)$ be a 2 by 2 real symmetric matrix, and $I=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ be the unit matrix.

1. Find all the eigenvalues ${\lambda }_1,{\lambda }_2$ of $M$.

2. Under the assumption of ${\lambda }_1\ne {\lambda }_2$, answer i) and ii). i) Let $U=(v_1,v_2)$ be the matrix whose first and second columns consist of the eigenvectors $v_1$ and $v_2$ for the eigenvalues ${\lambda }_1$ and ${\lambda }_2$, respectively. Show that $U$ is invertible and satisfies $M=U\left(\begin{array}{cc}{\lambda }_1& 0\\ 0& {\lambda }_2\end{array}\right)U^{-1}$. ii) Prove that the set $\left\{U^{-1}x|x\in Z^{\ast }\right\}$ and $Z^{\ast }$ are equal.

3. For each of the statements A), B), and C), answer the conditions on matrix elements $a,b,d$ for the statement to hold. A) Every $y\in Z^{\ast }$ can be expressed as $y=Mx$ with some $x\in Z^{\ast }$. B) No $y\in Z^{\ast }$ can be expressed as $y=Mx$ with some $x\in Z^{\ast }$. C) At least one $y\in Z^{\ast }$ can be expressed as $y=(M-{\lambda }_{1}I)x$ with some $x\in Z^{\ast }$.

正确解答

1. Finding the Eigenvalues

To find the eigenvalues of the matrix $M=\left(\begin{array}{cc}a& b\\ b& d\end{array}\right)$, we solve the characteristic equation:

$$\det (M-\lambda I)=0$$

The characteristic polynomial of $M$ is:

$$\det \left(\begin{array}{cc}a-\lambda & b\\ b& d-\lambda \end{array}\right)=(a-\lambda )(d-\lambda )-b^2=0$$

This simplifies to:

$${\lambda }^2-(a+d)\lambda +(ad-b^2)=0$$

The eigenvalues ${\lambda }_1$ and ${\lambda }_2$ are the roots of this quadratic equation:

$${\lambda }_{1,2}=\frac{(a+d)\pm \sqrt{(a+d{)}^2-4(ad-b^2)}}{2}$$

2. i) Showing $U$ is Invertible

Let $v_1$ and $v_2$ be the eigenvectors corresponding to ${\lambda }_1$ and ${\lambda }_2$, respectively. Define the matrix $U=(v_1,v_2)$. Since ${\lambda }_1\ne {\lambda }_2$, the eigenvectors $v_1$ and $v_2$ are linearly independent, and thus $U$ is invertible.

To show that $M=U\left(\begin{array}{cc}{\lambda }_1& 0\\ 0& {\lambda }_2\end{array}\right)U^{-1}$, consider the action of $M$ on the eigenvectors:

$$M{v}_1={\lambda }_1{v}_1\text{and}M{v}_2={\lambda }_2{v}_2$$

Therefore,

$$M(v_1,v_2)=(M{v}_1,M{v}_2)=({\lambda }_1{v}_1,{\lambda }_2{v}_2)=(v_1,v_2)\left(\begin{array}{cc}{\lambda }_1& 0\\ 0& {\lambda }_2\end{array}\right)$$

Thus, we have:

$$M=U\left(\begin{array}{cc}{\lambda }_1& 0\\ 0& {\lambda }_2\end{array}\right)U^{-1}$$

ii) Proving Set Equality

To prove that the set $\left\{U^{-1}x|x\in Z^{\ast }\right\}$ and $Z^{\ast }$ are equal, consider any $x\in Z^{\ast }$. Then $U^{-1}x\in Z^{\ast }$ if and only if $|z_1{|}^2+|z_2{|}^2\ne 0$. Since $U$ is invertible and $Z^{\ast }$ consists of all non-zero complex vectors, applying $U^{-1}$ to any vector in $Z^{\ast }$ yields another non-zero complex vector, ensuring the sets are equal.

3. Conditions for Statements

A) For every $y\in Z^{\ast }$ to be expressible as $y=Mx$ for some $x\in Z^{\ast }$, $M$ must be invertible. This requires ${\lambda }_1\ne 0$ and ${\lambda }_2\ne 0$, ensuring $a\ne 0$, $d\ne 0$, and $ad-b^2\ne 0$.

B) No $y\in Z^{\ast }$ can be expressed as $y=Mx$ for some $x\in Z^{\ast }$ if $M$ is singular and its image does not cover $Z^{\ast }$. This happens when $M$ has a zero eigenvalue, i.e., $ad-b^2=0$ and one of the eigenvalues is zero.

C) At least one $y\in Z^{\ast }$ can be expressed as $y=(M-{\lambda }_{1}I)x$ for some $x\in Z^{\ast }$ if $a=d$ and $b=0$ do not both hold true. This requires $M-{\lambda }_{1}I$ to be invertible or have a non-trivial image, which is true if ${\lambda }_1$ is not an eigenvalue of $M$, ensuring ${\lambda }_2\ne {\lambda }_1$.

知识点

特征值特征向量矩阵分解

解题技巧和信息

1. 特征值问题中，特征多项式是重要的工具，通过求解特征多项式可以得到特征值。
2. 当矩阵的特征值不同时，其特征向量是线性无关的，这使得特征向量矩阵是可逆的。
3. 在处理复杂矩阵时，注意到特征向量的规范性及其在不同基底下的表示。

重点词汇

eigenvalue 特征值

eigenvector 特征向量

invertible 可逆的

characteristic polynomial 特征多项式

quadratic equation 二次方程

参考资料

1. 《线性代数及其应用》 第 5 章 特征值和特征向量