Exercise 6.8.15

Answers

1.

A

is a square diagonal matrix, then we have

A_{ij} = A_{ji} = 0

2.

A

is a matrix congruent to a diagonal matrix

B

, then we have

B = Q^{t} AQ

and

A = {(Q^{- 1})}^{t} B Q^{- 1}

. This means

A

is symmetric since

A^{t} = {(Q^{- 1})}^{t} B^{t} Q^{- 1} = {(Q^{- 1})}^{t} B Q^{- 1} = A .

3.

Say

α

to be the standard basis and

β

to be the basis in Theorem 6.35. Let

H = ψ_{α}^{- 1} (A)

be the bilinear form whose matrix representation is

A

. Thus we know that

ψ_{α} (H) = A

and

ψ_{β} (H)

are congruent. Also, by Theorem 6.35 we know that

ψ_{β} (H)

is diagonal.

jephianlin

2011-06-27 00:00