if T: V -> 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.
The Hint that it gives is the equality:
T^2 - (lambda)^2 * I = (T + lambda * I)*(T - lambda * I)
where I is the identity matrix.
Any help?
By hypothesis,
equivalently
T^2-\lambda^2 I)(x)=0" alt="\exists x\in V\;(x\neq 0)T^2-\lambda^2 I)(x)=0" />.
Now, use:
and analyze what would happen if
.
Fernando Revilla