Find the maximum value of \(Q\left( {\rm{x}} \right) = 7x_1^2 + 3x_2^2 - 2x_1^{}x_2^{}\), subject to the constraint \(x_1^2 + x_2^2 = 1\). (Do not go on to find a vector where the maximum is attended.)

As per the question, we have:

\(Q\left( {\rm{x}} \right) = 7x_1^2 + 3x_2^2 - 2x_1^{}x_2^{}\)

This implies that corresponding matrix is\(A = \left[ {\begin{array}{*{20}{c}}7&{ - 1}\\{ - 1}&3\end{array}} \right]\).

Find the eigenvalues by solving \(\det \left( {A - \lambda I} \right) = 0\).

\(\begin{array}{c}\det \left( {\left[ {\begin{array}{*{20}{c}}{7 - \lambda }&{ - 1}\\{ - 1}&{3 - \lambda }\end{array}} \right]} \right) = 0\\\left( {7 - \lambda } \right)\left( {3 - \lambda } \right) - \left( { - 1} \right)\left( { - 1} \right) = 0\\21 - 7\lambda - 3\lambda + {\lambda ^2} - 1 = 0\\{\lambda ^2} - 10\lambda + 20 = 0\\\lambda = \frac{{10 \pm \sqrt {100 - 80} }}{2}\\ = 5 \pm \sqrt 5 \end{array}\)

Thus, the eigenvalues are \({\lambda _1} = 5 + \sqrt 5 \) and \({\lambda _2} = 5 - \sqrt 5 \).

The maximum value of the given function subjected to constraints\(x_1^2 + x_2^2 = 1\)will be the greatest eigenvalue.

So, we have:

\({\lambda _1} = 5 + \sqrt 5 \)

Hence, the required maximum value is at: \({\lambda _1} = 5 + \sqrt 5 \).