Question

What is the largest value of the quadratic form \({\bf{5}}x_{\bf{1}}^{\bf{2}}{\bf{ - 3}}x_{\bf{2}}^{\bf{2}}\) if \({{\bf{x}}^T}{\bf{x = 1}}\)?

Short answer

The largest possible value of the given quadratic form is \(5\).

Step-by-step solution

Step 1: Find the coefficient matrix of the given quadratic form

Consider the quadratic form \(5x_1^2 - 3x_2^2\),

\(\begin{aligned}{}5x_1^2 - 3x_2^2 &= \left( {\begin{aligned}{{}}{{x_1}}&{{x_2}}\end{aligned}} \right)\left( {\begin{aligned}{{}}5&0\\0&{ - 3}\end{aligned}} \right)\left( {\begin{aligned}{{}}{{x_1}}\\{{x_2}}\end{aligned}} \right)\\ &= {{\bf{x}}^T}A{\bf{x}}\end{aligned}\)

Therefore, thecoefficient matrix of the quadratic form is\(A = \left( {\begin{aligned}{{}}5&0\\0&{ - 3}\end{aligned}} \right)\).

Step 2: Find the maximum value

As from the diagonal matrix, the eigenvalues are \(5\) and \( - 3\).

Thus, the largest possible value of the given quadratic form is \(5\).