Lecture 8 : Eigenvalues , Eigenvectors and Spectral Theorem

Proof. Suppose Mv = λv. We want to show that λ has imaginary value 0. For a complex number x = a + ib, the conjugate of x, is defined as follows: x∗ = a − ib. So, all we need to show is that λ = λ∗. The conjugate of a vector is the conjugate of all of its coordinate. Taking the conjugate transpose of both sides of the above equality, we have v∗M = λ∗v∗, (8.1) where we used that M = M . So, on one hand, v∗Mv = v∗(Mv) = v∗(λv) = λ(v∗v). and on the other hand, by (8.1) v∗Mv = λ∗v∗v. So, we must have λ = λ∗.