九州大学 システム情報科学府 情報理工学専攻・電気電子工学専攻 2024年1月実施 線形代数

Author

祭音Myyura (assisted by ChatGPT 5.4 Thinking)

Description

Let be an real matrix and be an -dimensional nonzero real column vector. Define

where is the transpose of a vector . Answer the following questions.

(1) For

find .

(2) Find all the eigenvalues of . For each eigenvalue of , find its eigenspace.

(3) Suppose that is a symmetric matrix. Show

where is the largest eigenvalue of . Use the fact that the symmetric matrix has eigenvectors that form an orthonormal basis of .

Kai

(1)

We are given

By definition,

First compute :

Then

Also,

Therefore,

Hence,

(2)

To find the eigenvalues, solve

We have

Thus,

Now compute the determinant:

Since

we get

Therefore,

So the eigenvalues are

The eigenvalue has algebraic multiplicity , and the eigenvalue has algebraic multiplicity .

Eigenspace for

Solve

We have

Let

Then the system is

Hence,

Therefore,

So the eigenspace corresponding to is

Eigenspace for

Solve

We have

Let

The system gives

Thus,

Here and are free variables. Therefore,

So the eigenspace corresponding to is

(3)

Suppose is a symmetric matrix.

Since is symmetric, it has an orthonormal basis of eigenvectors

for .

Let their corresponding eigenvalues be

That is,

Let

Since form an orthonormal basis, any nonzero vector can be written as

Equivalently,

Because , we have

Now compute :

By linearity,

Since

we get

Next,

Expanding this expression gives

Because the vectors are orthonormal,

Therefore, all cross terms vanish, and we obtain

Similarly,

Thus,

Using orthonormality again, we get

Therefore,

Since

we have

Therefore,

Hence,

Since

we may divide both sides by . This gives

So for every nonzero vector ,

Thus,

Now we show that this upper bound is achieved.

Let be an eigenvector corresponding to the largest eigenvalue . Then

Choose

Then

Using

we get

Thus,

Since

we have

Therefore, the maximum value is actually attained, and we conclude that