eigenvalues of a Hermitian matrix are real

The eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath of a Hermitian (or self-adjoint) matrix are real.


Suppose λ is an eigenvalue of the self-adjoint matrix A with non-zero eigenvectorMathworldPlanetmathPlanetmathPlanetmath v. Then Av=λv.


Since v is non-zero by assumptionPlanetmathPlanetmath, vHv is non-zero as well and so λ*=λ, meaning that λ is real. ∎

Title eigenvalues of a Hermitian matrix are real
Canonical name EigenvaluesOfAHermitianMatrixAreReal
Date of creation 2013-03-22 14:23:09
Last modified on 2013-03-22 14:23:09
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 8
Author Andrea Ambrosio (7332)
Entry type Theorem
Classification msc 15A57