Exercise 2210Linear algebra › Quadratic formsL1Added 3 Nov 2026

Minimizing a quadratic form on the sphere

#minimum#sphere#eigenvalues

Statement

Minimize the quadratic form
$Q(x,y)=3x^2+2xy+3y^2$
on the unit circle $x^2+y^2=1$ .

Hint

The minimum is the smallest eigenvalue of the matrix $A=\begin{pmatrix}3&1\\1&3\end{pmatrix}$ .

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Matrix. $A=\begin{pmatrix}3&1\\1&3\end{pmatrix}$ .

Eigenvalues. $\det(A-\lambda I)=(3-\lambda)^2-1$ , roots $\lambda=2$ and $\lambda=4$ .

Rayleigh quotient. For $\|X\|=1$ , $X^TAX$ ranges in $[\lambda_{\min},\lambda_{\max}]=[2,4]$ .

Minimum. The minimum is $2$ , attained at the unit eigenvector for $\lambda=2$ , i.e. $(1,-1)/\sqrt2$ .

Conclusion. $\min Q=2$ on the unit circle, attained at $\pm(1,-1)/\sqrt2$ .