I'm stuck on the second part of the verification, and i can't seem to figure out how to find the eigenvalues.
Verify that $f(z) = \overline{z}$ is a linear operator. Find eigenvalues for f, for each eigenvalue find an eigenvector.
1
$\begingroup$
linear-algebra
linear-transformations
1 Answers
2
Since $f^2$ is the identity map, all the eigenvalues are $1$ or $-1$. Eigenvectors with eigenvalue $1$ are those elements that are unchanged by $f$, so we see that any element on the real line $x+0i$ is an eigenvector with eigenvalue $1$. For $-1$, the eigenvectors are the imaginary line $0+yi$, since $f(yi)=-yi$ for all real $y$.