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 2022年8月実施 問題8

Author

zephyr

Description

(1) Describe the eigenvalues and eigenvectors of the following matrix. ( is a real value.)

(2) What is the range of such that is positive semidefinite.

(3) Consider an symmetric matrix where all diagonal elements are and all non-diagonal elements are . Show that this matrix is non-singular when .

Kai

(1) Eigenvalues and Eigenvectors

To find the eigenvalues of the matrix , we need to solve the characteristic equation .

Using the row addition method for simplification, we can add all rows to the first row:

Now, we can absorb the common factor from the first row:

Setting the determinant to zero:

So, the eigenvalues are:

Eigenvectors

For :

Solving

, we find:

For :

We need to solve , which gives the eigenvectors corresponding to . This generally results in two linearly independent eigenvectors orthogonal to :

(2) Positive Semi-definiteness

A matrix is positive semidefinite if all its eigenvalues are non-negative. For to be positive semidefinite:

Solving these inequalities:

The most restrictive condition is .

Thus, the range of such that is positive semidefinite is:

(3) Non-Singularity of Symmetric Matrix

Consider an symmetric matrix where all diagonal elements are and all non-diagonal elements are .

The matrix can be written as:

where is the matrix with all elements equal to 1.

The eigenvalues of are (with multiplicity 1) and (with multiplicity ). Thus, the eigenvalues of are:

The matrix is non-singular if all its eigenvalues are non-zero:

This holds if:

since implies when .

Thus, the matrix is non-singular when .

Knowledge

特征值和特征向量 正定矩阵

难点解题思路

1. 通过求解特征方程来找到特征值。
2. 根据特征值的符号判断矩阵的半正定性。
3. 使用矩阵特征值的性质判断矩阵的非奇异性。

解题技巧和信息

1. 计算特征方程时，使用行列式和代数余子式。
2. 确定半正定矩阵时，所有特征值必须为非负数。
3. 判断矩阵是否非奇异，可以通过特征值是否全非零来实现。

重点词汇

eigenvalue 特征值

eigenvector 特征向量

positive semidefinite 正半定

non-singular 非奇异

参考资料

1. Linear Algebra and Its Applications by Gilbert Strang, Chap. 6
2. Introduction to Linear Algebra by Gilbert Strang, Chap. 7