Smallest value of a quadratic form

MediumLinear Algebra~4m

Let $A=\begin{pmatrix}3&1\\1&3\end{pmatrix}$ . As the unit vector $x$ (with $\|x\|=1$ ) ranges over all directions, the quadratic form $x^{\top}Ax$ takes a range of values. What is the smallest value it can take?

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Over unit vectors, $x^{\top}Ax$ is bounded by the eigenvalues of $A$ .
Find $\lambda_{\min}$ from $\lambda^2-6\lambda+8=0$ .

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Smallest value of a quadratic form

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