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Linear Algebra

Exercise 6.8.24

Answers

1.

Compute that

H (a x_{1} + x_{2}, y) = ⟨ a x_{1} + x_{2}, T (y) ⟩

= a ⟨ x_{1}, T (y) ⟩ + ⟨ x_{2}, T (y) ⟩

= aH (x_{1}, y) + H (x_{2}, y)

and

H (x, a y_{1} + y_{2}) = ⟨ x, T (a y_{1} + y_{2}) ⟩ = ⟨ x, aT (x_{1}) + T (x_{2}) ⟩

= a ⟨ x, T (y_{1}) ⟩ + ⟨ x, T (y_{2}) ⟩

= aH (x, y_{1}) + H (x, y_{2}) .

2.

Compute that

H (y, x) = ⟨ y, T (x) ⟩

= ⟨ T (x), y ⟩ = ⟨ x, T^{∗} (y) ⟩ .

The value equal to $H (x, y) = ⟨ x, T (y) ⟩$ for all $x$ and $y$ if and only if $T = T^{∗}$ .

3.

By Exercise 6.4.22 the operator

T

must be a positive semidefinite operator.

4.

It fail since

H (x, iy) = ⟨ x, T (iy) ⟩ = ⟨ x, iT (y) ⟩

= - i ⟨ x, T (y) ⟩ \neq iH (x, y)

in general.

jephianlin

2011-06-27 00:00

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