The question in the image asks for the gradient vector and Hessian matrix of the quadratic form:

$f(x) = x^T Q x$

where $x$ is a vector and $Q$ is a matrix.

Solution:

1. Gradient Vector: The gradient of the quadratic form $f(x)$ is calculated by differentiating with respect to the vector $x$. Since $x^T Q x$ is a quadratic expression, the gradient is:

   $\nabla f(x) = 2 Q x$

   This assumes that $Q$ is symmetric. If $Q$ is not symmetric, you would use the symmetrized version, $\frac{Q + Q^T}{2}$.

2. Hessian Matrix: The Hessian matrix is the matrix of second-order partial derivatives. Differentiating the gradient vector again with respect to $x$ yields the Hessian matrix:

   $H(f(x)) = 2 Q$

   Again, if $Q$ is not symmetric, the Hessian would be $2 \cdot \frac{Q + Q^T}{2}$.

Summary:

• Gradient vector: $\nabla f(x) = 2 Q x$
• Hessian matrix: $H(f(x)) = 2 Q$

Would you like more details on this? Let me know if you need further clarification!

Here are some related questions you may find interesting:

1. How do you derive the gradient of a quadratic form in general?
2. What are the conditions under which the Hessian matrix of a quadratic form is positive definite?
3. What is the significance of the Hessian matrix in optimization problems?
4. How do you handle non-symmetric matrices when computing the gradient and Hessian?
5. Can you use the same process for cubic forms or higher-order polynomial forms?

Tip: In optimization problems, the sign and definiteness of the Hessian matrix tell you about the nature of critical points (minima, maxima, or saddle points).