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 2016年8月実施 問題8

Author

zephyr

Description

Answer the following questions about linear algebra.

(1) Denote by the zero vector. Let denote a two-dimensional vector that is not . is the orthogonal projection of a point on . Prove the following propositions.

• (1.1) for any two-dimensional point .

• (1.2) for any non-zero two-dimensional vector orthogonal to .

(2) Assume that a real symmetric matrix satisfies . Prove that the eigenvalues of are either 0 or 1.

(3) Denote by the column vectors corresponding to the bases of a two-dimensional subspace of the three dimensional space. Describe the projection matrix to the subspace using .

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回答以下有关线性代数的问题。

(1) 用 表示零向量。设 表示一个二维向量，它不是 。 是点 在 上的正交投影。证明以下命题。

• (1.1) 对于任意二维点 ，。

• (1.2) 对于任意非零二维向量 ，它与 正交，。

(2) 假设一个实对称矩阵 满足 。证明 的特征值要么是 0，要么是 1。

(3) 用 表示对应于三维空间的二维子空间基的列向量。用 描述该子空间的投影矩阵。

Kai

(1)

(1.1)

The orthogonal projection of on is given by:

To prove the proposition, we apply again on :

Using the definition of orthogonal projection:

Simplifying the dot products:

Thus:

(1.2)

Given , we need to show:

Using the definition of orthogonal projection:

Since :

Thus:

(2)

Assume that a real symmetric matrix satisfies . Prove that the eigenvalues of are either 0 or 1.

Let be a real symmetric matrix. Therefore, it is diagonalizable. Let be an eigenvector of with eigenvalue :

Applying again:

Since , we have:

Thus:

Since is a non-zero vector, we can conclude:

Thus, the eigenvalues must satisfy:

Therefore:

(3)

Given the matrix formed by two column vectors and , which represent the basis of a two-dimensional subspace in three-dimensional space, we want to find the projection matrix that projects any vector in onto this subspace.

Matrix is:

where is a matrix.

Derivation of the Projection Matrix

1. Projection of a Vector

The projection of a vector onto the subspace spanned by the columns of can be expressed as a linear combination of the columns of :

In matrix form, we write:

where is a column vector of coefficients:

2. Finding the Coefficients

To determine the coefficients , we use the property that the projection minimizes the distance to the subspace. This can be formulated as:

This equation implies:

Assuming is invertible, we solve for :

3. Constructing the Projection Matrix

Substituting back into the projection formula, we have:

Since this holds for any vector , the projection matrix can be identified as:

Knowledge

对称矩阵 特征值和特征向量 投影矩阵

重点词汇

• Orthogonal projection 正交投影
• Symmetric matrix 对称矩阵
• Eigenvalue 特征值
• Column vector 列向量
• Subspace 子空间
• Projection matrix 投影矩阵

参考资料

1. Gilbert Strang, "Linear Algebra and Its Applications," Chap. 3, 5.
2. David C. Lay, "Linear Algebra and Its Applications," Chap. 6.