跳到主要内容

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 2019年8月実施 問題8

Author

zephyr

Description

Assume that the following equation holds for square matrices, , and :

Assume that , is the inverse matrix of , , and if .

Solve the following problems.

(1) Show the inverse matrix for each of and .

(2) Show that is one of the eigenvalues of , and show one of the corresponding eigenvectors of .

(3) Suppose that is a positive integer. Show every pair of eigenvalue and corresponding eigenvector of .


假设对于 的方阵 ,以下等式成立:

假设 的逆矩阵,,且当

解决以下问题。

(1) 展示 的逆矩阵。

(2) 证明 的一个特征值,并展示 的一个对应特征向量。

(3) 假设 是一个正整数。展示 的每对特征值和对应特征向量。

Kai

(1)

Since is a diagonal matrix with diagonal entries , its inverse, denoted as , is also a diagonal matrix. The diagonal entries of are the reciprocals of the diagonal entries of :

To find the inverse matrix of , use the given equation . Multiply both sides by and appropriately:

Taking the inverse of both sides, we get:

Using the property of inverses for matrix products:

Thus, is given by:

(2)

Given the equation , this implies that is the diagonal form of under the similarity transformation by . The diagonal entries of , denoted as , are the eigenvalues of .

To show this formally, consider , where is the -th standard basis vector. We have:

Since , let . Then:

Hence, is an eigenvector of corresponding to the eigenvalue .

(3)

From the similarity transformation , raising both sides to the power gives:

Since :

Because is diagonal, is also diagonal, with each diagonal element being raised to the power :

Thus, has the same eigenvectors as , and the eigenvalues are the -th powers of the eigenvalues of . Therefore, the eigenvalue-eigenvector pairs for are:

  • Eigenvalue:
  • Corresponding eigenvector:

In summary, every eigenvalue of raised to the power is an eigenvalue of , and the corresponding eigenvectors remain the same.

Knowledge

特征值和特征向量 相似变换

重点词汇

  • Matrix: 矩阵
  • Eigenvalue: 特征值
  • Eigenvector: 特征向量
  • Inverse matrix: 逆矩阵
  • Diagonal matrix: 对角矩阵
  • Similarity transformation: 相似变换

参考资料

  1. Axler, S. (2015). Linear Algebra Done Right. Springer. Chap. 8
  2. Strang, G. (2009). Introduction to Linear Algebra. Wellesley-Cambridge Press. Chap. 5