特征多项式
特征值而且特征向量是方阵的性质,并用于许多现实生活中的应用,包括几何变换和概率场景。为了找到这些特征值和特征向量特征多项式因为一个矩阵必须被找到并求解。
什么是特征多项式?
- 对于一个矩阵
" class="Wirisformula" role="math" alt="粗体斜体A" style="vertical-align:-6px;height:22px;width:16px">,如果A {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="加粗斜体x等于加粗斜体x" style="vertical-align:-6px;height:22px;width:63px">当A x = λ x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="粗体斜体x" style="vertical-align:-6px;height:22px;width:12px">是一个非零向量和x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="λ" style="vertical-align:-6px;height:22px;width:11px">一个常数,然后λ {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="λ" style="vertical-align:-6px;height:22px;width:11px">是一个特征值矩阵的λ {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="粗体斜体A" style="vertical-align:-6px;height:22px;width:16px">而且A {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="粗体斜体x" style="vertical-align:-6px;height:22px;width:12px">它是对应的特征向量x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 如果
" class="Wirisformula" role="math" alt="加粗斜体A加粗斜体x等于向右双箭头左括号加粗斜体I减去加粗斜体A右括号加粗斜体x等于0" style="vertical-align:-6px;height:22px;width:181px">或A x = λ x ⇒ ( λ I - A ) x = 0 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="左括号加粗斜体A - I右括号加粗斜体x = 0" style="vertical-align:-6px;height:22px;width:99px">和( A - λ I ) x = 0 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="粗体斜体x" style="vertical-align:-6px;height:22px;width:12px">要成为一个非零向量,x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="det空格左括号斜体加粗I减去斜体加粗A右括号等于0" style="vertical-align:-6px;height:22px;width:114px">det ( λ I - A ) = 0 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - an的特征多项式
" class="Wirisformula" role="math" alt="N叉乘N" style="vertical-align:-6px;height:22px;width:38px">矩阵:n × n {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 在这门课中,你只需要找到a的特征方程
" class="Wirisformula" role="math" alt="2叉乘2" style="vertical-align:-6px;height:22px;width:36px">矩阵,这个总是a二次2 × 2 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
我怎么找到特征多项式?
- 步骤1
写 " class="Wirisformula" role="math" alt="加粗斜体I减去加粗斜体A" style="vertical-align:-6px;height:22px;width:50px">记住单位矩阵的顺序必须和λ I - A {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="粗体斜体A" style="vertical-align:-6px;height:22px;width:16px">A {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 步骤2
求行列式 " class="Wirisformula" role="math" alt="加粗斜体I减去加粗斜体A" style="vertical-align:-6px;height:22px;width:50px">使用公式小册子上给你的公式λ I - A {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
- 步骤3
重写成一个多项式
考试技巧
- 你需要记住特征方程事实就是这样不分子式小册子上有
工作的例子
求下列矩阵的特征多项式
特征值和特征向量
你怎么找到一个矩阵的特征值?
- 矩阵的特征值
" class="Wirisformula" role="math" alt="粗体斜体A" style="vertical-align:-6px;height:22px;width:16px" loading="lazy">都是通过解特征多项式矩阵的A {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 在这门课中,因为特征多项式总是a二次时,多项式总是会生成以下其中之一:
- 两个真实的和截然不同的特征值,
- 一个真实的重复特征值或
- 复杂的特征值
怎么找到特征v载体矩阵的?
- 的值
" class="Wirisformula" role="math" alt="粗体斜体x" style="vertical-align:-6px;height:22px;width:12px" loading="lazy">满足方程的是an特征向量的矩阵x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="粗体斜体A" style="vertical-align:-6px;height:22px;width:16px" loading="lazy">A {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 任意标量的倍数
" class="Wirisformula" role="math" alt="粗体斜体x" style="vertical-align:-6px;height:22px;width:12px" loading="lazy">也满足方程,因此有无数对应于特定特征值的特征向量x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 步骤1
写 " class="Wirisformula" role="math" alt="粗体斜体x等于开括号表x行y结束表闭括号" style="vertical-align:-21px;height:54px;width:61px" loading="lazy">x = x y {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 步骤2
把特征值代入方程 " class="Wirisformula" role="math" alt="左括号斜体加粗I减去斜体加粗A右括号斜体加粗x = 0" style="vertical-align:-6px;height:22px;width:99px" loading="lazy">,形成两个方程( λ I - A ) x = 0 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="x" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">而且x {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} " class="Wirisformula" role="math" alt="y" style="vertical-align:-6px;height:22px;width:11px" loading="lazy">y {"language":"en","fontFamily":"Times New Roman","fontSize":"18"} - 步骤3
这个方程会有无穷多个解,所以让其中一个变量等于来选择一个解 " class="Wirisformula" role="math" alt="1" style="vertical-align:-6px;height:22px;width:10px" loading="lazy">然后用它来求另一个变量1 {"language":"en","fontFamily":"Times New Roman","fontSize":"18"}
考试技巧
- 你可以快速检查一下计算出来的特征值主对角线你所分析的矩阵应该总和到特征值的总和对于矩阵
工作的例子
求下列矩阵的特征值和相关特征向量。
一)
b)