(实对称矩阵的相似对角化)
数学
线性代数
实对称矩阵的相似对角化
实对称矩阵的性质
在之前我们有学过，若 $A^T = A$ ，则 $A$ 为对称矩阵，进一步，若组成 $A$ 的元素都是实数，则 $A$ 为实对称矩阵.
- $A$ 是是对称矩阵，则 $A$ 的特征值为实数，特征向量为实向量
- 实对称矩阵 $A$ 的属于不同特征值的特征向量相互正交
- 对任意的n阶实对称矩阵 $A$ ，存在n阶正交矩阵 $Q$ ，使得：
$$ Q^TAQ = Q^{-1}AQ = \begin{bmatrix} \lambda_1 &&& \ & \lambda_2 && \ && \ddots & \ &&& \lambda_n \end{bmatrix} $$
这里 $\lambda_i(1 \leq i \leq n)$ 是 $A$ 的全部特征值.
实对称矩阵相似对角化的基本步骤
若 $A$ 为n阶实对称矩阵，则其用正交矩阵 $Q$ 相似对角化的基本步骤如下：
- 求 $A$ 的特征值 $\lambda_1, \lambda_2, \cdots, \lambda_n$
- 求 $A$ 的对应于特征值 $\lambda_1, \lambda_2, \cdots, \lambda_n$ 的特征向量 $\xi_1, \xi_2, \cdots, \xi_n$
- 将 $\xi_1, \xi_2, \cdots, \xi_n$ 正交化（若需要的话），单位化为 $\eta_1, \eta_2, \cdots, \eta_n$
- 令 $Q = [\eta_1, \eta_2, \cdots, \eta_n]$ ，则 $Q$ 为正交矩阵，且 $Q^{-1}AQ = Q^TAQ = B$
英语
每日一句
From career to community and family, these contrasts suggest that in the aftermath of the searing Great Recession, those just starting out in life are defining priorities and expectations that will increasingly spread through virtually all aspects of American life, from consumer preferences to housing patterns to politics.(2016, Reading Comprehension, Part A Text 4)
词汇
contrast: n. 差异，差别；对比，对照 aftermath: n. （战争、风暴、事故的）余波，后果 the Great Recession: 经济大萧条，大衰退 searing: adj. 灼热的；灼痛的，剧痛的 define: v. 下定义；解释，界定 definition: n. 定义；解释 expectation: n. 期望，预料；指望 expect: v. 预料，期望；认为 spread through: 蔓延到…… virtually: adv. 几乎，实际上，差不多 consumer preference: 消费者偏好 housing pattern: 居住模式
第一步：找谓语
From career to community and family, these contrasts ==suggest== that in the aftermath of the searing Great Recession, those just starting out in life ==are defining== priorities and expectations that ==will== increasingly ==spread== through virtually all aspects of American life, from consumer preferences to housing patterns to politics.
第二步：断句
原句中存在3处谓语，包含3件事，其谓语分别分布在以下位置：
- suggest 为主句谓语
- are defining 为 that 引导的宾语从句谓语
- will spread 为 that 引导的定语从句谓语
按照标点、引导词以及谓语，可以将原句断开为以下分句：
- From career to community and family, these contrasts ==suggest== —— 主句
- that in the aftermath of the searing Great Recession, those just starting out in life ==are defining== priorities and expectations —— 宾语从句
- that ==will== increasingly ==spread== through virtually all aspects of American life, from consumer preferences to housing patterns to politics. —— 定语从句
第三步：简化
主句
From career to community and family, these contrasts suggest
- 介词短语：From career to community and family, 为范围状语，修饰主句
- 该部分为：from A to B and C 译为：从A到B和C
- 主句主语部分：these contrasts
- 限定词：these 修饰名词：contrasts
- 名词：contrasts 为主语核心词
- 主句谓语部分：suggest 为及物动词，后接宾语
去掉主句扩展部分，就得到了主句核心：
- …… contrasts suggest —— …… 对比建议
宾语从句
that in the aftermath of the searing Great Recession, those just starting out in life are defining priorities and expectations
- 从句引导词：that 引导宾语从句，作动词：suggest 的宾语
- 介词短语：in the aftermath of the searing Great Recession, 为时间状语，修饰从句，说明从句发生的时间
- 定冠词：the 修饰名词：aftermath
- 名词：aftermath 为介词：in 的宾语
- 介词短语：of the searing Great Recession 为名词：aftermath 的后置定语，说明具体的余波为经济大萧条
- 从句主语部分：those just starting out in life
- 副词：just 修饰动词：starting
- 非谓语动词短语：starting out in life 为后置定语，解释说明代词：those
- 介词短语：in life 为介宾短语作范围状语，修饰动词：starting
- 从句谓语部分：are defining 为及物动词的现在进行时态，后接宾语
- 从句宾语部分：priorities and expectations 为名词：priorities 与名词：expectations 由并列连词：and 连接而成的并列结构，共同作为动词：are defining 的宾语
去掉从句的扩展成分，就得到了从句核心：
- that …… those …… are defining priorities and expectations —— …… 那些 …… 正在定义的优先事项与期望
定语从句
that will increasingly spread through virtually all aspects of American life, from consumer preferences to housing patterns to politics.
- 从句引导词：that 引导定语从句，作为名词词组：priorities and expectations 的限制性定语从句，解释说明是哪种优先事项与期望
- 引导词：that 在从句中作主语
- 从句谓语部分：will increasingly spread
- 副词：increasingly 修饰动词：will spread
- 动词：will spread 为不及物动词的将来时态
- 介词短语：through virtually all aspects of American life, from consumer preferences to housing patterns to politics. 为范围状语，修饰动词：will spread 说明传播的具体范围
- 副词：virtually 修饰形容词：all
- 形容词：all 修饰名词：aspects
- 名词：aspects 为介词：through 的宾语
- 介词短语：of American life 为后置定语，修饰：aspects ，说明 aspects 的具体范围
- 介词短语：from consumer preferences to housing patterns to politics 为同位语，例举说明：aspects of American life
去掉从句的扩展部分，就得到了从句核心：
- that will …… spread …… —— 优先事项与期望将会 …… 传播 ……
