Monday, 28 September 2015

linear algebra - If an endomorphism satisfies alpha=alpha, then its eigenvalues are purely imaginary



This is another exercise from Golan's book.



Problem: Let V be an inner product space over C and let α be an endomorphism of V satisfying α=α, where α denotes the adjoint. Show that every eigenvalue of α is purely imaginary.



My proposed solution is below.


Answer




Let me show another argument which applies to a more general setting: if α is a linear operator on a Hilbert space satisfying α=α, then the spectrum of α is purely imaginary (i.e. real part equal zero).



Indeed, one simply needs to notice that αλid is invertible if and only if (αλid) is invertible. As (αλid)=α¯λid=α¯λid=(α+¯λid), we conclude that any λ in the spectrum of α satisfies ¯λ=λ.


No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find lim without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...