Question

What is the largest possible value of the quadratic form \({\bf{5}}x_{\bf{1}}^{\bf{2}} + {\bf{8}}x_{\bf{2}}^{\bf{2}}\) if \({\bf{x}} = \left( {{x_{\bf{1}}},{x_{\bf{2}}}} \right)\) and \({{\bf{x}}^T}{\bf{x = 1}}\), that is, if \(x_{\bf{1}}^{\bf{2}} + x_{\bf{2}}^{\bf{2}} = {\bf{1}}\)? (Try some examples of \({\bf{x}}\).

Short answer

The largest possible value of the given quadratic form is \(8\).

Step-by-step solution

Find the matrix form of the given quadratic form

Consider the quadratic form \(5x_1^2 + 8x_2^2\),

\(\begin{aligned}{}5x_1^2 + 8x_2^2 &= \left( {\begin{aligned}{{}}{{x_1}}&{{x_2}}\end{aligned}} \right)\left( {\begin{aligned}{{}}5&0\\0&8\end{aligned}} \right)\left( {\begin{aligned}{{}}{{x_1}}\\{{x_2}}\end{aligned}} \right)\\ &= {{\bf{x}}^T}A{\bf{x}}\end{aligned}\)

Step 2: Find the eigen vector

As from the diagonal matrix, the eigenvalues are \(5\) and \(8\). Therefore, eigenvectors are shown below:

\(\lambda = 8:\left( {\begin{aligned}{{}}0\\{ \pm 1}\end{aligned}} \right)\)and \(\lambda = 5:\left( {\begin{aligned}{{}}{ \pm 1}\\0\end{aligned}} \right)\)

Since the vectors are in normalized form thus, the largest possible value of the given quadratic form is \(8\).