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Exercise 5.1.17

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1.

T (A) = A^{t} = λA

for some

λ

and some nonzero matrix

A

, say

A_{ij} \neq 0

, we have

A_{ij} = λ A_{ji}

and

A_{ji} = λ A_{ij}

and so

A_{ij} = λ^{2} A_{ij} .

This means that $λ$ can only be $1$ or $- 1$ . And these two values are eigenvalues due to the existence of symmetric matrices and skew-symmetric matrices.

2.

The set of nonzero symmetric matrices are the eigenvectors corresponding to eigenvalue

1

, while the set of nonzero skew-symmetric matrices are the eigenvectors corresponding to eigenvalue

- 1

3.

It could be

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

4.

Let

E_{ij}

be the matrix with its

ij

-entry

1

and all other entries

0

. Then the basis could be

{E_{ii}}_{i = 1, 2, \dots, n} \cup {E_{ij} + E_{ji}}_{i > j} \cup {E_{ij} - E_{ji}}_{i > j} .

jephianlin

2011-06-27 00:00

Exercise 5.1.17

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