 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-1/5/i/solution

H^{†} = H

, where

†

v

with

H v = λ v

v^{†} H v

equals its own complex conjugate, and

λ = \frac{v ^{†} H v}{v ^{†} v} \in R .

Thus every Hermitian eigenvalue is real.

U^{†} U = U U^{†} = I

. If

Uv = μv

, preservation of the squared norm gives

v^{†} v = (Uv)^{†} (Uv) = ∣ μ ∣^{2} v^{†} v .

Since

v \neq = 0

∣ μ ∣ = 1

. These statements concern complex eigenvectors, even when the matrix entries happen to be real.

 New to topics? Read the docs here!