The given problem is about finding the gradient of a function $f(x)$ and determining if the solution to $\nabla f(x) = 0$ will be unique.

Problem Statement Recap:

Let $f(x) = \frac{1}{2}\|Ax - b\|^2 + \|x\|^2$, where $A \in \mathbb{R}^{3 \times 4}$ and $b \in \mathbb{R}^3$. The matrices $A$ and $b$ are defined as:

$A = \begin{bmatrix} 3 & 2 & 0 & -1 \\ -1 & 3 & 0 & 2 \\ 0 & -4 & -2 & 7 \end{bmatrix}, \quad b = \begin{bmatrix} 3 \\ 1 \\ -4 \end{bmatrix}$

We need to find the gradient of $f$ with respect to $x = [x_1, x_2, x_3, x_4]^T$ and determine whether the solution to $\nabla f(x) = 0$ will be unique.

Step 1: Finding the Gradient $\nabla f(x)$

The function $f(x)$ can be written as:

$f(x) = \frac{1}{2} \|Ax - b\|^2 + \|x\|^2$

Using the Euclidean norm properties, we have:

$\|Ax - b\|^2 = (Ax - b)^T (Ax - b)$ $\|x\|^2 = x^T x$

Thus, $f(x)$ becomes:

$f(x) = \frac{1}{2}(Ax - b)^T (Ax - b) + x^T x$

The gradient of $f(x)$ with respect to $x$ is:

$\nabla f(x) = A^T (Ax - b) + 2x$

Step 2: Analyzing Uniqueness of Solution

To check whether the solution to $\nabla f(x) = 0$ is unique, we rewrite the condition for optimality:

$A^T (Ax - b) + 2x = 0$

This can be rearranged as:

$(A^T A + 2I)x = A^T b$

Here, $I$ is the $4 \times 4$ identity matrix. The uniqueness of the solution depends on whether $(A^T A + 2I)$ is invertible. For $(A^T A + 2I)$ to be invertible, it must be a positive definite matrix.

• $A^T A$ is a $4 \times 4$ matrix, and adding $2I$ makes it $(A^T A + 2I)$ positive definite, since adding a positive definite matrix to a positive semi-definite matrix results in a positive definite matrix.
• A positive definite matrix is always invertible.

Thus, $(A^T A + 2I)$ being positive definite implies it is invertible. Therefore, the solution $x$ to the equation $(A^T A + 2I)x = A^T b$ will be unique.

Conclusion

Without finding the actual values of $x$, we can conclude that the solution to $\nabla f(x) = 0$ is unique because the matrix $(A^T A + 2I)$ is positive definite and hence invertible.

Do you want more details or have any questions?

Here are five related questions that expand on the previous information:

1. What are the eigenvalues of the matrix $(A^T A + 2I)$, and how do they relate to the matrix's positive definiteness?
2. Can we explicitly compute the solution $x$ using the normal equations derived?
3. How does the dimensionality of $A$ (being $3 \times 4$) affect the solution space of $Ax = b$ in general?
4. What would change if $A$ were a square matrix (i.e., $4 \times 4$ instead of $3 \times 4$)?
5. How would the problem formulation and solution change if we used a different norm (e.g., $L_1$ norm) instead of the Euclidean norm?

Tip: When dealing with optimization problems, checking the positive definiteness of the Hessian matrix or its equivalent ensures that the problem has a unique minimum and that any solution found is stable.