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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 2024年1月実施 問題8

Author​

zephyr

Description​

Suppose that the eigenvalues and the corresponding eigenvectors of an square matrix are and respectively.

Suppose that is the identity matrix, and the inverse matrix of an invertible matrix is .

Answer the following questions.

  1. Show all the eigenvalues and the corresponding eigenvectors of .

  2. If are mutually different, show that is a diagonal matrix, using that is a matrix of concatenated eigenvectors.

  3. Show all the eigenvalues and the corresponding eigenvectors of .

  1. Suppose that is the maximum eigenvalue of , and . Calculate .

  2. Suppose that is the eigenvector of corresponding to the minimum eigenvalue. Calculate using that is a matrix concatenating .

  3. Suppose that is an arbitrary positive integer. Calculate .

假设 方阵 的特征值及相应的特征向量分别为 和 。

假设 是 的单位矩阵，并且可逆矩阵 的逆矩阵为 。

回答以下问题。

  1. 展示 的所有特征值及相应的特征向量。

  2. 如果 是互不相同的，证明 是一个对角矩阵，其中 是由特征向量构成的矩阵。

  3. 展示 的所有特征值及相应的特征向量。

  1. 假设 是 的最大特征值，并且 。
    计算 。

  2. 假设 是 对应于最小特征值的特征向量。计算 ，其中 是由 构成的矩阵。

  3. 假设 是任意正整数。计算 。

Kai​

1. Positive Eigenvalues and Normalized Eigenvectors of ​

Given the singular value decomposition (SVD) of as , we can express as follows:

The matrix is diagonal with the diagonal elements (). Thus, the positive eigenvalues of are exactly the , and the associated normalized eigenvectors are the columns of .

2. Surjectivity and Injectivity of ​

Surjective (onto): The mapping is surjective if the range of spans , i.e., has full row rank. This occurs when .

Injective (one-to-one): The mapping is injective if the kernel of contains only the zero vector, i.e., has full column rank. This occurs when .

3. Image of and Kernel of ​

The pseudoinverse is defined as . Consider .

We need to show that is isomorphic to . Observe the following:

Thus, .

Now, consider . Then , and

Thus, . Therefore, .

4. Orthogonal Decomposition​

Given where and :

To show orthogonality:

Since is symmetric ():

Thus, and are orthogonal.

5. Minimizing ​

Let . We need to show that minimizes the expression.

Consider the error:

Since , we have , thus:

The norm to be minimized is:

This is minimized when since and .

Knowledge​

重点词汇​

  • singular value decomposition (SVD) 奇异值分解
  • pseudoinverse 广义逆
  • surjective 满射
  • injective 单射
  • orthogonal decomposition 正交分解

参考资料​

  1. "Linear Algebra and Its Applications" by Gilbert Strang, Chapter 7: The Singular Value Decomposition (SVD)
  2. "Matrix Computations" by Gene H. Golub and Charles F. Van Loan, Chapter 2: Matrix Analysis
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