京都大学 情報学研究科 通信情報システム専攻 2024年8月実施 専門基礎A [A-1]

Author

祭音Myyura

Description

(1)

(a) Evaluate the following integral:

(b) Using the result of Question (a), evaluate the following integral:

(c) Gamma function is defined as follows:

Using the result from Question (b), find the value of .

(2)

Matrix is given as:

(a) Find all the eigenvalues and their corresponding eigenvectors of matrix .

(b) Let be a positive integer. Find .

Kai

(1)

(a)

The integral can be solved using polar coordinates, where and . The integral becomes:

We use a substitution , .

The inner integral evaluates to .

(2)

The integral can be related to by noting that .

Since , we have . As the integrand is positive, must be positive. .

(c)

and with , we get

(2)

(a)

We find the eigenvalues by solving the characteristic equation , which yields . The roots are , , .

We solve for each eigenvalue to find the corresponding eigenvectors:

• For , the eigenvector is proportional to .
• For , the eigenvector is proportional to .
• For , the eigenvector is proportional to .

(b)

The matrix is diagonalizable as , where is the matrix of eigenvectors and is the diagonal matrix of eigenvalues. .

Hence The matrix is: