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 2023年8月実施 問題8

Author

zephyr

Description

Answer the following questions regarding the transition matrix

,

.

(1) Show that the vector

is an eigenvector of .

(2) Find the -th order transition matrix .

(3) Find .

The trace of an real-valued matrix is defined as . Answer the following questions.

(4) Prove that where the eigenvalues of are described as .

(5) Prove the following: If there exists a natural number such that ( is the zero matrix), then .

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回答以下关于转移矩阵

的问题，

。

(1) 证明向量

是 的一个特征向量。

(2) 求 阶转移矩阵 。

(3) 求 。

一个 实数值矩阵 的迹定义为 。回答以下问题。

(4) 证明 ，其中 的特征值为 。

(5) 证明以下命题：如果存在自然数 使得 （ 是零矩阵），则 。

Kai

Written by zephyr

解题思路

这道题目涉及马尔可夫链的转移矩阵、特征向量、矩阵幂运算、极限计算以及矩阵迹的性质。我们需要运用线性代数的知识来分析转移矩阵的特征值和特征向量，利用这些来计算矩阵的幂和极限。最后，我们还需要证明关于矩阵迹的一些性质。

Solution

1. Eigenvector Proof

To show that is an eigenvector of , we need to find a scalar such that .

Let's compute:

Therefore, is indeed an eigenvector of with eigenvalue .

2. n-th Order Transition Matrix

To find , we'll use the eigen-decomposition of . We already found one eigenpair. Let's find the other:

The characteristic equation is:

Solving this, we get: and

The eigenvector corresponding to is

Now we can write , where:

Therefore,

3. Limit of as

As , since . Therefore,

4. Proof of

Let be diagonalizable (if not, we can use the Jordan canonical form). Then , where is a diagonal matrix with eigenvalues on the diagonal.

5. Proof: If , then

We will prove this statement using the following steps:

1. Let be the eigenvalues of (including algebraic multiplicities).

2. From the condition , we know that is nilpotent. For a nilpotent matrix, all its eigenvalues are zero. We can prove this as follows:

   If is an eigenvalue of , then is an eigenvalue of . Since , all eigenvalues of must be zero. Therefore, , which implies .

3. From the result in part 4, we know that .

4. Since all eigenvalues of are zero, we have:

Therefore, we have proved that if , then .

Knowledge

难点思路

本题的难点在于第2问和第3问，需要利用矩阵的特征分解来计算矩阵的幂。关键是要认识到可以用特征值和特征向量将矩阵对角化，从而简化矩阵幂的计算。另外，在计算矩阵的逆时要特别注意，因为这里很容易出错。

解题技巧和信息

1. 在处理马尔可夫链问题时，注意转移矩阵的特征值总有一个是1。
2. 计算矩阵的高次幂时，考虑使用特征分解方法。
3. 在证明矩阵性质时，考虑使用特征值的性质。
4. 矩阵迹的性质在很多证明中都很有用，要熟练掌握。
5. 在进行矩阵运算时，要仔细检查每一步，特别是在计算逆矩阵时。

重点词汇

• Markov chain 马尔可夫链
• transition matrix 转移矩阵
• eigenvalue 特征值
• eigenvector 特征向量
• matrix diagonalization 矩阵对角化
• trace of a matrix 矩阵的迹
• characteristic equation 特征方程
• steady state 稳态

参考资料

1. Strang, Gilbert. "Linear Algebra and Its Applications." Chapter 5: Eigenvalues and Eigenvectors.
2. Ross, Sheldon M. "Introduction to Probability Models." Chapter 4: Markov Chains.
3. Horn, Roger A., and Charles R. Johnson. "Matrix analysis." Chapter 1: Eigenvalues, Eigenvectors, and Similarity.