We are tasked with finding the derivative of the function $f(A) = \|Ab\|^2$ with respect to the matrix $A$, where $b = [1, \cdots, 1]^T \in \mathbb{R}^{n \times 1}$ and $A \in \mathbb{R}^{n \times n}$. The Euclidean norm is used, which means:

$\|Ab\|^2 = (Ab)^T(Ab)$

Step 1: Express the function explicitly

The norm squared can be written as:

$f(A) = \|Ab\|^2 = (Ab)^T(Ab) = b^T A^T A b$

Here, $A^T$ is the transpose of $A$, and $b$ is a column vector of ones in $\mathbb{R}^{n \times 1}$.

Step 2: Differentiate with respect to $A$

To compute the derivative of $f(A)$ with respect to $A$, we can use matrix calculus. Recall the following result for differentiating a quadratic form:

$\frac{\partial}{\partial A} \left( x^T A^T A x \right) = 2 A x x^T$

Now, in our case, we have $x = b$. Therefore, the derivative of $f(A)$ with respect to $A$ is:

$\frac{\partial f}{\partial A} = 2 A b b^T$

Conclusion

Thus, the derivative of $f(A)$ with respect to $A$ is:

$\frac{\partial f}{\partial A} = 2 A b b^T$

This result shows how the function $f(A)$ changes with respect to the matrix $A$.

Do you need further clarification or details on this process?

Here are five related questions that could help expand on this topic:

1. How would the derivative change if we used a different norm, such as the $L_1$-norm or the $L_\infty$-norm?
2. What is the impact of the vector $b$ on the gradient $\frac{\partial f}{\partial A}$?
3. How would this derivative be used in an optimization context, for example, when minimizing $f(A)$?
4. Can we compute the second derivative of $f(A)$ with respect to $A$, and what would it look like?
5. How do matrix calculus rules change when dealing with non-square matrices or other types of vector norms?

Tip: When working with matrix calculus, it's helpful to remember standard derivative formulas, such as the one used above for quadratic forms, as they simplify many matrix optimization problems.