MHB Q(x) is a strictly convex function, show that G is positive definite

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The discussion centers on proving that the symmetric matrix G is positive definite if the quadratic function q(x) is strictly convex. The participants explore the relationship between strict convexity and the properties of the gradient and Hessian, noting that strict convexity implies any stationary point is a unique global minimizer. A key insight is that the strict convexity of q(x) leads to the inequality q(0) < (1/2)(q(x) + q(-x)), which implies that x^T G x > 0 for any nonzero vector x. This conclusion confirms that G must be positive definite, demonstrating the connection between the properties of the quadratic function and the matrix G.
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Consider the quadratic function $\displaystyle q(\textbf{x}) = \frac{1}{2} \textbf{x}^T G \textbf{x} + \textbf{d}^T \textbf{x} + c$ on $\mathbb{R}^n$, where $\textbf{d}$ is a constant $n \times 1$ vector, $G$ is a constant $n \times n$ symmetric matrix and $c$ is a scalar.

The gradient is $\nabla q(\textbf{x}) = G \textbf{x} + \textbf{d}$ and the Hessian is $\nabla^2 q(\textbf{x}) = G$.

If $q(\textbf{x})$ is a strictly convex function then show that $G$ is positive definite.I am not sure whether I should start with the convex function definition or start by considering the gradient or the Hessian.

I tried expanding the inequality in the convex function definition but didn't get anywhere.

There is a proposition that says $f$ is strictly convext on $\mathbb{R}^n$ $\implies$ any stationary point is the unique global minimizer. (I can't even prove that a stationary point exists) Another theorem says that positive definiteness is a sufficient condition for being a unique global minimizer and positive semi definiteness is a necessary condition for being a local minimizer. I can't see how to use these statements to prove what the question is asking.
 
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numbersense said:
Consider the quadratic function $\displaystyle q(\textbf{x}) = \frac{1}{2} \textbf{x}^T G \textbf{x} + \textbf{d}^T \textbf{x} + c$ on $\mathbb{R}^n$, where $\textbf{d}$ is a constant $n \times 1$ vector, $G$ is a constant $n \times n$ symmetric matrix and $c$ is a scalar.

The gradient is $\nabla q(\textbf{x}) = G \textbf{x} + \textbf{d}$ and the Hessian is $\nabla^2 q(\textbf{x}) = G$.

If $q(\textbf{x})$ is a strictly convex function then show that $G$ is positive definite.I am not sure whether I should start with the convex function definition or start by considering the gradient or the Hessian.

I tried expanding the inequality in the convex function definition but didn't get anywhere.

There is a proposition that says $f$ is strictly convext on $\mathbb{R}^n$ $\implies$ any stationary point is the unique global minimizer. (I can't even prove that a stationary point exists) Another theorem says that positive definiteness is a sufficient condition for being a unique global minimizer and positive semi definiteness is a necessary condition for being a local minimizer. I can't see how to use these statements to prove what the question is asking.
I don't think you need to use the gradient or the Hessian to show that $G$ is positive definite. The fact that $q$ is strictly convex tells you that $q(\textbf{0}) <\frac12\bigl(q(\textbf{x}) + q(-\textbf{x})\bigr)$, for any nonzero vector $\textbf{x}.$ It follows very easily that $0 < \textbf{x}^T G \textbf{x}$.
 
Opalg said:
I don't think you need to use the gradient or the Hessian to show that $G$ is positive definite. The fact that $q$ is strictly convex tells you that $q(\textbf{0}) <\frac12\bigl(q(\textbf{x}) + q(-\textbf{x})\bigr)$, for any nonzero vector $\textbf{x}.$ It follows very easily that $0 < \textbf{x}^T G \textbf{x}$.

Thanks. It follows like this.
\begin{align*}
q\left(\frac{1}{2} \textbf{x} + \left(1 - \frac{1}{2}\right) (-\textbf{x})\right) = q(0) = c <& \frac{1}{2} q(\textbf{x}) + \left(1 - \frac{1}{2}\right) q(-\textbf{x}) = \textbf{x}^T G \textbf{x} + c\\
\textbf{x}^T G\textbf{x} >& 0
\end{align*}

This is quite clever. I didn't think of obtaining an inequality by using specific values in the convex function definition.
 
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