Exercise 7.2.1

Answers

1.

Yes. A diagonal matrix is a Jordan canonical form of itself. And the Corollary after Theorem 7.10 tells us that the Jordan canonical form is unique.

2.

Yes. This is a result of Theorem 7.11.

3.

No. The two matrices

(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})

and

(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})

have different Jordan canonical form. By Theorem 7.11, they cannot be similar.

4.

Yes. This is a result of Theorem 7.11.

5.

Yes. They are just two matrix representations of one transformation with different bases.

6.

No. The two matrices

(\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix})

and

(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})

have different Jordan canonical form.

7.

No. The identity mapping

I

from

ℂ^{2}

ℂ^{2}

has two different bases

{(1, 0), (0, 1)}

and

{(1, 1), (1, - 1)} .

8.

Yes. This is the Corollary after Theorem 7.10.

jephianlin

2011-06-27 00:00