Certify a quadratic form

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Problem

Consider $q(x,y)=2x^2-2xy+2y^2$ . Write $q=z^{\mathsf T}Hz$ , determine whether $H$ is positive definite, and find the minimum of $q$ over $x^2+y^2=1$ .

Reveal answer or reference solution

The symmetric matrix is

H=\begin{bmatrix}2&-1\\-1&2\end{bmatrix}

Its eigenvalues are $1$ and $3$ , both positive, so $H$ is positive definite. On the unit circle the Rayleigh quotient is minimized by an eigenvector for the smallest eigenvalue. Hence the minimum is $1$ , attained at $z=\pm(1,1)^{\mathsf T}/\sqrt2$ .

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Certify a quadratic form

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