京都大学 情報学研究科 通信情報システム専攻 2024年2月実施 専門基礎A [A-1]

Author

祭音Myyura

Description

(1)

Find the limit

(2)

Evaluate the integral

(3)

Given the matrix:

(i) Find the eigenvalues and corresponding eigenvectors of , assuming it has distinct eigenvalues.

(ii) Using the eigenvalues and eigenvectors from (i), find , where is a positive integer.

(iii) If the eigenvalues are the same, find for a positive integer .

Kai

(1)

We know the Maclaurin series for is:

Thus,

As , the limit is:

(2)

We express and in terms of and : , . The integrand becomes . The Jacobian of the transformation is calculated as .

The integral in the -coordinate system is given by:

We integrate with respect to first, then :

Evaluating the final integral:

(3)

(i)

The characteristic equation is:

The determinant is:

Thus, the eigenvalues are and .

The corresponding eigenvectors are:

• For , the eigenvector is .
• For , the eigenvector is .

(ii)

If has distinct eigenvalues, it is diagonalizable. We can write , where is the diagonal matrix of eigenvalues and is the matrix whose columns are the corresponding eigenvectors:

The inverse of is .Then :

(iii)

If , then is not diagonalizable, and we use the Jordan form to find :