東京大学 新領域創成科学研究科 メディカル情報生命専攻 2015年8月実施 問題8

标签：

• Tokyo-University
• Matrix-Inverse
• Covariance-Matrix
• Eigenvalues-and-Eigenvectors

Author

zephyr, 祭音Myyura

Description

Answer the following questions about linear algebra.

(1)

Compute the inverse matrix of the following matrix,

$$\begin{pmatrix} 1 & 2 \\ 2 & 5 \end{pmatrix}.$$

(2)

Consider data points $(x_i, y_i), i = 1, \dots, n$ in a two-dimensional space. Variance with respect to the x-axis, variance with respect to the y-axis, and covariance are respectively defined as

$$\sigma_x = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2, \quad \sigma_y = \frac{1}{n} \sum_{i=1}^{n} (y_i - \bar{y})^2, \quad \sigma_{xy} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})$$

where $\bar{x}, \bar{y}$ denote the averages with respect to the x and y axes, respectively.

A: Compute the variance-covariance matrix

$$\begin{pmatrix} \sigma_x & \sigma_{xy} \\ \sigma_{xy} & \sigma_y \end{pmatrix}$$

for the following data points, $(-2, -2), (2, 2), (1, -1), (-1, 1)$.

B: Compute all eigenvalues and eigenvectors of the variance-covariance matrix.

(3)

Prove that, if the eigenvalues of a regular matrix A are $\lambda_1, \dots, \lambda_n$, those of the inverse matrix A$^{-1}$ are $1/\lambda_1, \dots, 1/\lambda_n$.

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回答以下关于线性代数的问题。

(1)

计算以下矩阵的逆矩阵，

$$\begin{pmatrix} 1 & 2 \\ 2 & 5 \end{pmatrix}.$$

(2)

考虑数据点 $(x_i, y_i), i = 1, \dots, n$ 在二维空间中。相对于 x 轴的方差、相对于 y 轴的方差和协方差分别定义为

$$\sigma_x = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2, \quad \sigma_y = \frac{1}{n} \sum_{i=1}^{n} (y_i - \bar{y})^2, \quad \sigma_{xy} = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})$$

其中 $\bar{x}, \bar{y}$ 分别表示相对于 x 和 y 轴的平均值。

A: 计算方差-协方差矩阵

$$\begin{pmatrix} \sigma_x & \sigma_{xy} \\ \sigma_{xy} & \sigma_y \end{pmatrix}$$

对于以下数据点，$(-2, -2), (2, 2), (1, 1), (-1, 1)$。

B: 计算方差-协方差矩阵的所有特征值和特征向量。

(3)

证明，如果一个正规矩阵 A 的特征值是 $\lambda_1, \dots, \lambda_n$，那么其逆矩阵 A$^{-1}$ 的特征值是 $1/\lambda_1, \dots, 1/\lambda_n$。

题目描述

回答下列线性代数问题：

1. 求矩阵

   $$\begin{pmatrix}1&2\\2&5\end{pmatrix}$$

   的逆矩阵。

2. 对二维数据点 $(x_i,y_i)$（$i=1,\ldots,n$），定义

   $$\sigma_x=\frac1n\sum_{i=1}^{n}(x_i-\bar x)^2,\qquad \sigma_y=\frac1n\sum_{i=1}^{n}(y_i-\bar y)^2,$$
   $$\sigma_{xy}=\frac1n\sum_{i=1}^{n}(x_i-\bar x)(y_i-\bar y),$$

   其中 $\bar x,\bar y$ 为两坐标的均值。

   1. 对数据点

      $$(-2,-2),\ (2,2),\ (1,-1),\ (-1,1)$$

      求方差—协方差矩阵

      $$\begin{pmatrix}\sigma_x&\sigma_{xy}\\\sigma_{xy}&\sigma_y\end{pmatrix}.$$

   2. 求该方差—协方差矩阵的全部特征值与特征向量。

3. 证明：若可逆矩阵 $\mathbf A$ 的特征值为 $\lambda_1,\ldots,\lambda_n$，则 $\mathbf A^{-1}$ 的相应特征值为 $1/\lambda_1,\ldots,1/\lambda_n$。

注：Description 的英文题干及 Kai 均给出第三个数据点为 $(1,-1)$；原有中文译文在此处写成了 $(1,1)$，以上按两处一致的信息重组。

Kai

(1)

To find the inverse of the matrix

$$\mathbf{A} = \begin{pmatrix} 1 & 2 \\ 2 & 5 \end{pmatrix},$$

we use the formula for the inverse of a $2 \times 2$ matrix:

$$\mathbf{A}^{-1} = \frac{1}{\det(\mathbf{A})} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix},$$

where $\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ and $\det(\mathbf{A}) = ad - bc$.

For our matrix,

$$a = 1, \quad b = 2, \quad c = 2, \quad d = 5.$$

First, compute the determinant:

$$\det(\mathbf{A}) = (1 \cdot 5) - (2 \cdot 2) = 5 - 4 = 1.$$

Then, the inverse is

$$\mathbf{A}^{-1} = \frac{1}{1} \begin{pmatrix} 5 & -2 \\ -2 & 1 \end{pmatrix} = \begin{pmatrix} 5 & -2 \\ -2 & 1 \end{pmatrix}.$$

(2)

A: Variance-Covariance Matrix

Given data points $(-2, -2), (2, 2), (1, -1), (-1, 1)$, we first compute the mean values:

$$\bar{x} = \frac{-2 + 2 + 1 - 1}{4} = 0, \quad \bar{y} = \frac{-2 + 2 - 1 + 1}{4} = 0.$$

Next, we compute the variances and covariances:

$$\sigma_{xx} = \frac{1}{4} \sum_{i=1}^4 (x_i - \bar{x})^2 = \frac{1}{4} [(2)^2 + (2)^2 + (1)^2 + (1)^2] = \frac{10}{4} = 2.5,$$

$$\sigma_{yy} = \frac{1}{4} \sum_{i=1}^4 (y_i - \bar{y})^2 = \frac{1}{4} [(2)^2 + (2)^2 + (1)^2 + (1)^2] = \frac{10}{4} = 2.5,$$

$$\sigma_{xy} = \frac{1}{4} \sum_{i=1}^4 (x_i - \bar{x})(y_i - \bar{y}) = \frac{1}{4} [(-2)(-2) + (2)(2) + (1)(-1) + (-1)(1)] = \frac{6}{4} = 1.5.$$

Thus, the variance-covariance matrix is:

$$\mathbf{C} = \begin{pmatrix} 2.5 & 1.5 \\ 1.5 & 2.5 \end{pmatrix}.$$

B: Eigenvalues and Eigenvectors

To find the eigenvalues $\lambda$ of $\mathbf{C}$, solve the characteristic equation:

$$\det(\mathbf{C} - \lambda \mathbf{I}) = 0.$$

For our matrix $\mathbf{C}$,

$$\mathbf{C} - \lambda \mathbf{I} = \begin{pmatrix} 2.5 - \lambda & 1.5 \\ 1.5 & 2.5 - \lambda \end{pmatrix},$$

the determinant is:

$$\det(\mathbf{C} - \lambda \mathbf{I}) = (2.5 - \lambda)(2.5 - \lambda) - (1.5)(1.5) = \lambda^2 - 5\lambda + 4 = 0.$$

Solving for $\lambda$, we get:

$$\lambda^2 - 5\lambda + 4 = 0 \implies (\lambda - 4)(\lambda - 1) = 0 \implies \lambda_1 = 4, \quad \lambda_2 = 1.$$

To find the eigenvectors corresponding to the eigenvalues $\lambda_1 = 4$ and $\lambda_2 = 1$, we solve the equation $(\mathbf{C} - \lambda \mathbf{I})\mathbf{v} = \mathbf{0}$.

For $\lambda_1 = 4$

$$\mathbf{C} - 4\mathbf{I} = \begin{pmatrix} 2.5 - 4 & 1.5 \\ 1.5 & 2.5 - 4 \end{pmatrix} = \begin{pmatrix} -1.5 & 1.5 \\ 1.5 & -1.5 \end{pmatrix}.$$

The equation $(\mathbf{C} - 4\mathbf{I})\mathbf{v} = \mathbf{0}$ becomes:

$$\begin{pmatrix} -1.5 & 1.5 \\ 1.5 & -1.5 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}.$$

This gives us the system of equations:

$$-1.5v_1 + 1.5v_2 = 0, \\ 1.5v_1 - 1.5v_2 = 0.$$

From the first equation, we obtain $v_1 = v_2$. Therefore, an eigenvector corresponding to $\lambda_1 = 4$ is:

$$\mathbf{v}_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}.$$

For $\lambda_2 = 1$

$$\mathbf{C} - 1\mathbf{I} = \begin{pmatrix} 2.5 - 1 & 1.5 \\ 1.5 & 2.5 - 1 \end{pmatrix} = \begin{pmatrix} 1.5 & 1.5 \\ 1.5 & 1.5 \end{pmatrix}.$$

The equation $(\mathbf{C} - 1\mathbf{I})\mathbf{v} = \mathbf{0}$ becomes:

$$\begin{pmatrix} 1.5 & 1.5 \\ 1.5 & 1.5 \end{pmatrix} \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}.$$

This gives us the system of equations:

$$1.5v_1 + 1.5v_2 = 0, \\ 1.5v_1 + 1.5v_2 = 0.$$

From the first equation, we obtain $v_1 = -v_2$. Therefore, an eigenvector corresponding to $\lambda_2 = 1$ is:

$$\mathbf{v}_2 = \begin{pmatrix} 1 \\ -1 \end{pmatrix}.$$

Thus, all eigenvectors corresponding to $\lambda_1=4$ are $c\begin{pmatrix}1\\1\end{pmatrix}$ with $c\ne0$, and all eigenvectors corresponding to $\lambda_2=1$ are $c\begin{pmatrix}1\\-1\end{pmatrix}$ with $c\ne0$.

(3)

Let $\mathbf{A}$ be a regular matrix with eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_n$ and corresponding eigenvectors $\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n$. By definition, we have:

$$\mathbf{A} \mathbf{v}_i = \lambda_i \mathbf{v}_i \implies \mathbf{A}^{-1} \mathbf{A} \mathbf{v}_i = \mathbf{A}^{-1} (\lambda_i \mathbf{v}_i) \implies \mathbf{v}_i = \lambda_i \mathbf{A}^{-1} \mathbf{v}_i \implies \mathbf{A}^{-1} \mathbf{v}_i = \frac{1}{\lambda_i} \mathbf{v}_i.$$

Since $\mathbf A$ is invertible, every $\lambda_i$ is nonzero. Conversely, an eigenvalue $\mu$ of $\mathbf A^{-1}$ gives the eigenvalue $1/\mu$ of $\mathbf A$ by the same argument. Thus, the eigenvalues of $\mathbf{A}^{-1}$ are exactly $\frac{1}{\lambda_i}$ for $i = 1, 2, \ldots, n$.

Knowledge

矩阵逆 方差协方差矩阵 特征值和特征向量

解题技巧和信息

1. 计算逆矩阵时，确保熟记 $2 \times 2$ 矩阵的逆矩阵公式。
2. 计算方差-协方差矩阵时，需准确计算均值、方差和协方差。
3. 找特征值和特征向量时，熟悉特征值方程和特征向量的计算方法。
4. 证明部分注意利用特征值和特征向量的定义和性质。

重点词汇

• Inverse matrix: 逆矩阵
• Variance-Covariance matrix: 方差-协方差矩阵
• Eigenvalue: 特征值
• Eigenvector: 特征向量

参考资料

1. Gilbert Strang, Linear Algebra and Its Applications, Chapter 3.
2. Axler, Sheldon, Linear Algebra Done Right, Chapter 5.