- #1

Hall

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- Homework Statement
- If ##T: V \to V## has the property that ##T^2## has a non-negative eigenvalue ##\lambda^2## , prove that at least one

of ##\lambda## or ##-\lambda## is an eigenvalue for T. [Hint: ##T^2 - \lambda^2 = (T + \lambda I)(T - \lambda I)## .]

- Relevant Equations
- ##T(x)= \lambda x##

The statement " If ##T: V \to V## has the property that ##T^2## has a non-negative eigenvalue ##\lambda^2##", means that there exists an ##x## in ##V## such that ## T^2 (x) = \lambda^2 x##.

If ##T(x) = \mu x##, we've have

$$

T [T(x)]= T ( \mu x)$$

$$

T^2 (x) = \mu^2 x$$

$$

\lambda ^2 = \mu ^2 \implies \mu = \lambda ~or~ -\lambda$$.

Is my solution correct? The only thing which is quite not very well is that I assumed that there exists an eigenvalue ##\mu## and an eigenvector ##x## in ##T##.

What the hint in the question wants me to do?

If ##T(x) = \mu x##, we've have

$$

T [T(x)]= T ( \mu x)$$

$$

T^2 (x) = \mu^2 x$$

$$

\lambda ^2 = \mu ^2 \implies \mu = \lambda ~or~ -\lambda$$.

Is my solution correct? The only thing which is quite not very well is that I assumed that there exists an eigenvalue ##\mu## and an eigenvector ##x## in ##T##.

What the hint in the question wants me to do?