Problem 2 - Analysis

Let $A$ be an $n \times n$ real matrix, $x=(x_1,x_2,\dots,x_n)^T$ be a nonzero $n$-dimensional real vector. Here, $A^T$ and $x^T$ denote the transposes of $A$ and $x$, respectively.

1.

Set $f(x)=x^T A x$. Compute the following first and second derivatives of $f$:

$$\bigtriangledown f(x) = \left( \frac{\partial f}{\partial x_1},\dots,\frac{\partial f}{\partial x_n}\right)^T$$

$$\bigtriangledown^2 f(x) = \left( \begin{tabular}{cccc} $\frac{\pa... ...$ & $\cdots$ & $\frac{\partial^2 f}{\partial x_n^2}$ \\ \end{tabular} \right)$$

2.

Set $g(x)=\frac{x^T \bigtriangledown f(x)}{x^T x}$. Compute $\bigtriangledown g(x)$.

3.

When does $\bigtriangledown g(x) = 0$ hold?