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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 2017年8月実施 問題8

Author​

zephyr

Description​

Let be an real matrix with positive rank . Such a matrix has a singular value decomposition , where and are , real matrices, respectively, and satisfy , (: unit matrix, : transpose of matrix ). is an real diagonal matrix whose diagonal elements () satisfy .

(1) Describe all the positive eigenvalues and associated normalized eigenvectors of matrix .

(2) Let be a linear mapping defined by . Describe the conditions on such that is surjective. Also, describe the conditions on such that is injective.

(3) The pseudoinverse of is defined by . Let and define linear mapping by . Show that image is linearly isomorphic to kernel (: dimensional zero vector).

(4) Show that (, ) is an orthogonal decomposition.

(5) For a given , let . Show that minimizes . (Hint: )

设 为一个 的实矩阵，且正秩为 。这样的矩阵有一个奇异值分解 ，其中 和 分别是 、 的实矩阵，并且满足 ，（： 单位矩阵，：矩阵 的转置）。 是一个 的实对角矩阵，其对角元素 （）满足 。

(1) 描述矩阵 的所有正特征值和相关的归一化特征向量。

(2) 令 为由 定义的线性映射。描述 的条件，使得 是满射。同时，描述 的条件，使得 是单射。

(3) 的伪逆定义为 。令 并定义线性映射 由 。证明 在线性上同构于 （： 维零向量）。

(4) 证明 （，）是一个正交分解。

(5) 对于给定的 ，令 。证明 最小化 。 （提示：）

Kai​

(1)​

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)​

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)​

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)​

Given where and :

To show orthogonality:

Since is symmetric ():

Thus, and are orthogonal.

(5)​

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​

奇异值分解 线性映射 广义逆矩阵 正交分解 线性代数

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