A unified version of Cauchy–Schwarz and Wielandt inequalities

Let A   be an n×nn×n positive definite symmetric real matrix with n   eigenvalues λ1,λ2,…,λnλ1,λ2,…,λn and let x and y   be two n×1n×1 vectors with the angle ψψ. This paper proves the following inequality|xTAy|2⩽maxi,jλicos2ψ2-λjsin2ψ2λicos2ψ2+λjsin2ψ22(xTAx)(yTAy).It is a unified version of the Cauchy–Schwarz inequality and the Wielandt one.