- #1
LeifEricson
- 11
- 0
Homework Statement
let [tex]T:V \to V[/tex] be a linear transformation which satisfies [tex]T^2 = \frac{1}{2} (T + T^*) [/tex] and is normal. Prove that [tex]T=T^*[/tex].
Homework Equations
The Attempt at a Solution
I think we should start like this:
Let [tex]\mathbf{A}=[T]_B[/tex] be the matrix representation of T in the orthonormal base [tex]B[/tex]. We look at A as a matrix in the Complex plane. If we prove that for every Eigenvalue, [tex]\lambda[/tex], it happens that [tex]\lambda \in \mathbb{R}[/tex], then [tex]T = T^*[/tex] by a theorem and we finished.
Now, be [tex]\mathbf{x}[/tex] an Eigenvector that satisfies [tex]\mathbf{Ax} = \lambda \mathbf{x}[/tex], then [tex]<\mathbf{Ax},\mathbf{x}> = \lambda ||\mathbf{x}||^2[/tex] and [tex] \lambda = \frac{<\mathbf{Ax},\mathbf{x}>}{||\mathbf{x}||^2}[/tex]. If only I could show that [tex]<\mathbf{Ax},\mathbf{x}>[/tex] is a real number, I solved it.