東京大学 情報理工学系研究科 コンピュータ科学専攻 2015年2月実施 問題1

Author

kainoj

Description

(1) Suppose that is a real symmetric matrix. Let and be an eigenvector and the corresponding eigenvalue of , respectively. Prove that is a real number.

(2) Suppose that is a real symmetric matrix. Let and be eigenvectors of ; and and be the eigenvalues of that correspond to and , respectively. Furthermore, assume that .

Prove that . Here denotes the inner product of the two vectors and .

(3) Let be a matrix

where , , and ( ) are real numbers such that the two vectors and are linearly independent. Let us define a function by

Prove that the equation determines a plane in the space.

(4) In the setting of Question (3), give three points that lie on the plane determined by .

Kai

(1)

Let such that and for some eigenvector and corresponding eigenvalue . Suppose that , i.e. for some . Let's start with :

Hence, = 0, which means .

(2)

Start with the first eigenpair:

Now fiddle with the second eigenpair:

If we subtract both result, we get:

Since , then . In other words, .

(3) - send help

This seems to be difficult. I succeeded in computing determinant using Laplace expansion wrt first column, but it gets hardcore later. Any ideas?

(4)

Give 3 points that lie on plane . When is determinant ? For example, when some columns are dependent. Let's make some columns dependent. Zum Beispiel, let's take:

Now first and second column are dependent, so . To get two others solutions, take and or simply multiply the first solution by some constants.