広島大学 先進理工系科学研究科 情報科学プログラム 2017年8月実施 専門科目I 問題1

Author

samparker, 祭音Myyura

Description

とする。

(1) を対称行列 と交代行列 の和 に分解せよ。ただし、 は、行列 の転置を表す。

(2) のすべての固有値と対応する固有空間を求めよ。

(3) のすべての固有値と対応する固有空間を求めよ。

(4) 一般に、実交代行列の固有値は または純虚数であることを示せ。

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Let .

(1) Decompose into the sum of the symmetric matrix () and the alternative matrix (). Here denotes the transpose of the matrix .

(2) Find all the eigenvalues of the symmetric matrix and a basis of the corresponding eigenspaces.

(3) Find all the eigenvalues of the alternative matrix and a basis of the corresponding eigenspaces.

(4) Show that the eigenvalues of the real alternative matrix are or purely imaginary numbers.

Kai

(1)

Let and . Then we have

Solving the equations we have

Hence

(2)

Eigenvalues

A basis of the corresponding eigenspaces

(3)

Eigenvalues

A basis of the corresponding eigenspaces

(4)

Let be an eigenvalue of and let be an eigenvector corresponding to the eigenvalue . That is, we have

Multiplying by from the left, we have

Note that the left hand side is the dot (inner) product of and . Since the dot product is commutative, we have

Since is skew-symmetric, we have . Substituting this into the above equality, we have

Taking conjugate of and use the fact that is real, we have

Thus, we have

Therefore comparing the left and right hand sides of (*) yields

Since is an eigenvector, it is nonzero by definition. Thus .

Hence we have

and this implies that is either or purely imaginary number.