Exercise 2209Linear algebra › Quadratic formsL1Added 3 Nov 2026

Maximizing a quadratic form on the sphere

#maximum#sphere#eigenvalues

Statement

Maximize the quadratic form
$Q(x,y)=3x^2+2xy+3y^2$
on the unit circle.

Hint

Use the largest eigenvalue.

Show solutionHide solution

Eigenvalues. From the previous exercise, $\lambda_1=2$ and $\lambda_2=4$ .

Maximum. On the unit sphere, $Q\le\lambda_2=4$ , with equality at the unit eigenvector for $\lambda_2$ , i.e. $(1,1)/\sqrt2$ .

Verification. $Q(1/\sqrt2,1/\sqrt2)=3/2+2/2+3/2=4$ . Correct.

Conclusion. $\max Q=4$ on the unit circle, attained at $\pm(1,1)/\sqrt2$ .