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 2018年8月実施 問題8

Author

zephyr

Description

If an real symmetric matrix satisfies condition for any non-zero -dimensional real column vector , is called a positive definite matrix ( represents the transpose of ). Answer the following questions.

(1) Show the diagonal elements of real positive definite matrix are positive.

(2) Show the eigenvalues of real symmetric matrix are real.

(3) Show the eigenvalues of positive definite matrix are positive.

(4) Let be the set of non-zero -dimensional real column vectors of unit length. Show is the largest eigenvalue of positive definite matrix .

(5) Suppose the eigenvectors of positive definite matrix are all different. Furthermore, suppose you know the largest eigenvalue and its associated eigenvector . Explain how to compute the second largest eigenvalue using and without computing the third largest or smaller eigenvalues.

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如果一个 实对称矩阵 满足对于任意非零 维实列向量 ，有 ，则称 为正定矩阵 ( 表示 的转置)。回答以下问题。

(1) 证明实正定矩阵 的对角元素 是正的。

(2) 证明实对称矩阵 的特征值是实数。

(3) 证明正定矩阵 的特征值是正的。

(4) 设 为非零 维实列向量 的单位长度集合。证明 是正定矩阵 的最大特征值。

(5) 假设正定矩阵 的特征向量都不相同。此外，假设你知道最大特征值 及其关联的特征向量 。解释如何在不计算第三大或更小的特征值的情况下，使用 和 计算第二大特征值 。

Kai

(1)

Consider the standard basis vector where the -th element is 1 and all other elements are 0. Then,

Since is positive definite, we have:

Thus,

(2)

For a real symmetric matrix , consider its eigenvalue equation:

where is an eigenvector and is the corresponding eigenvalue. Since is symmetric, we have:

for any vectors and . Setting , we get:

Since (as ), must be real.

(3)

Let be a positive definite matrix and be an eigenvalue with corresponding eigenvector :

Taking the transpose of , we have:

Since is positive definite, and . Thus,

(4)

Definitions and Setup

• Let be an real symmetric positive definite matrix.
• The set represents the set of unit vectors in .
• We aim to show that the maximum value of the quadratic form over the set is the largest eigenvalue of .

Step-by-Step Proof

1. Spectral Theorem Application: Since is a real symmetric matrix, by the spectral theorem, it can be diagonalized as:

where is an orthogonal matrix (i.e., ) and is a diagonal matrix containing the eigenvalues of , with since is positive definite.

1. Quadratic Form Transformation: For any unit vector (i.e., ), we can express in terms of the orthonormal basis formed by the columns of :

where is a unit vector () in . Substituting this into the quadratic form gives:

1. Maximization over Unit Vectors: The expression can be written as:

where are the components of the vector . Given that (since is a unit vector), we need to maximize the weighted sum of the eigenvalues with respect to .

1. Eigenvalue Maximization:

   To maximize , note that the maximum value occurs when and (because is the largest eigenvalue). Hence,

Therefore,

1. Conclusion: The maximum value of the quadratic form over the unit sphere is indeed the largest eigenvalue of the positive definite matrix .

(5)

Given:

• is a positive definite matrix.
• has distinct eigenvalues .
• The largest eigenvalue and its associated eigenvector are known.

We aim to find the second largest eigenvalue using and .

Orthogonal Projection Method

1. Orthogonality of Eigenvectors: Since is symmetric, its eigenvectors corresponding to distinct eigenvalues are orthogonal. Thus, is orthogonal to all other eigenvectors of .

2. Defining the Subspace Orthogonal to : Let be the subspace of consisting of vectors orthogonal to :

1. Rayleigh Quotient in the Subspace: The Rayleigh quotient for a vector is given by:

We need to maximize this quotient over the subspace .

1. Projection Method: To find , we consider the effect of and deflate by projecting it onto the subspace orthogonal to .

   Define the matrix as:

This matrix removes the influence of and . The matrix still has the same eigenvectors as , but the eigenvalue associated with is replaced with 0.

1. Finding : The second largest eigenvalue of will be the largest eigenvalue of in the subspace orthogonal to .

   To find , consider any vector in :

Here, is the projection of onto the subspace orthogonal to . Since is in , .

1. Maximizing the Rayleigh Quotient: The Rayleigh quotient in the subspace for the matrix becomes:

Since , we have , and thus:

Therefore, the maximum value of in the subspace gives :

1. Conclusion: By maximizing the Rayleigh quotient in the subspace orthogonal to the eigenvector associated with , we find the second largest eigenvalue .

Knowledge

正定矩阵 特征值和特征向量 Rayleigh商 正交性

难点思路

计算次大特征值的过程可能是一个难点，尤其是如何正确地使用正交性和 Rayleigh 商。

解题技巧和信息

• 正定矩阵的定义和性质非常重要，尤其是对特征值的影响。
• 计算特征值时，Rayleigh 商是一个有效工具。
• 正交性在分解和简化问题中非常有用。

重点词汇

• Positive definite matrix 正定矩阵
• Eigenvalue 特征值
• Eigenvector 特征向量
• Rayleigh quotient Rayleigh 商
• Orthogonal 正交的

参考资料

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