広島大学先進理工系科学研究科情報科学プログラム 2022年8月実施専門科目I 問題1

Author

祭音Myyura

(1) Find all the eigenvalues and the corresponding eigenvectors of the 2-dimensional square matrix .

(2) Let be real. Then find all the eigenvalues of the 2-dimensional real symmetric matrix: and show that the eigenvalues are real.

(3) Let be the eigenvalues of the real symmetric matrix . Then the matrix can be diagonalized by using the orthogonal matrix:

where denotes the transpose of the matrix . Express and using .

For the given matrix:

To find the eigenvalues and eigenvectors, we need to solve the characteristic equation:

the eigenvalues and eigenvectors are

For this matrix, the characteristic equation is:

This results in the quadratic equation:

Expanding this gives:

The eigenvalues are the roots of this equation,

which are real because the discriminant is non-negative:

Then we have

which can be simplified to

Hence we have

Substituting this result back into

we find:

similarly,