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

Author

zephyr

Description

Answer the following questions about linear algebra.

(1)

Compute the inverse matrix of the following matrix,

(2)

Consider data points 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

where denote the averages with respect to the x and y axes, respectively.

A: Compute the variance-covariance matrix

for the following data points, .

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

(3)

Prove that, if the eigenvalues of a regular matrix A are , those of the inverse matrix A are .

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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，

(2)

考虑数据点 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

其中 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

对于以下数据点，。

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 ，那么其逆矩阵 A 的特征值是 。

Kai

(1)

To find the inverse of the matrix

we use the formula for the inverse of a matrix:

where and .

For our matrix,

First, compute the determinant:

Then, the inverse is

(2)

A: Variance-Covariance Matrix

Given data points , we first compute the mean values:

Next, we compute the variances and covariances:

Thus, the variance-covariance matrix is:

B: Eigenvalues and Eigenvectors

To find the eigenvalues of , solve the characteristic equation:

For our matrix ,

the determinant is:

Solving for , we get:

To find the eigenvectors corresponding to the eigenvalues and , we solve the equation .

For

The equation becomes:

This gives us the system of equations:

From the first equation, we obtain . Therefore, an eigenvector corresponding to is:

For

The equation becomes:

This gives us the system of equations:

From the first equation, we obtain . Therefore, an eigenvector corresponding to is:

Thus, the eigenvectors corresponding to the eigenvalues and are and , respectively.

(3)

Let be a regular matrix with eigenvalues and corresponding eigenvectors . By definition, we have:

Thus, the eigenvalues of are for .

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 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.