Let's break down and prove the identity step by step:

Problem Recap:

You are asked to show that for a function $f: \mathbb{R}^m \to \mathbb{R}^m$, where $f(y) = \left(f^1(y^1, \dots, y^m), \dots, f^m(y^1, \dots, y^m)\right)^T$, the following identity holds:

$(f_y f )_y f = f^T f_{(yy)} f + f_y^2 f$

Notation Recap:

• $f_y$ is the Jacobian matrix of $f$, which has entries $(f_y)_{ij} = \frac{\partial f^i}{\partial y^j} =: f^i_j$.
• $f_{(yy)}$ is a matrix whose entries are the second derivatives $\frac{\partial^2 f^i}{\partial y^k \partial y^l} =: f^i_{kl}$.
• $f_y^2$ is the matrix product $f_y f_y$.
• $(f_y f)_y$ represents taking the derivative of the product $f_y f$ with respect to $y$.

Step 1: Compute $(f_y f)_y f$

1. Start by calculating the derivative of the product $f_y f$ with respect to $y$:

   $(f_y f)_y = \frac{\partial}{\partial y^k} \left( \sum_{j=1}^m \frac{\partial f^i}{\partial y^j} f^j \right)$

2. Applying the product rule to the above expression:

   $(f_y f)_y = \sum_{j=1}^m \frac{\partial^2 f^i}{\partial y^k \partial y^j} f^j + \sum_{j=1}^m \frac{\partial f^i}{\partial y^j} \frac{\partial f^j}{\partial y^k}$

   The first term represents the Hessian matrix multiplied by $f^j$, and the second term is the product of the Jacobian matrices:

   $(f_y f)_y = f^T f_{(yy)} + f_y^2$

Step 2: Multiply by $f$ on the Right

Now, multiply the above expression by $f$ from the right side:

$(f_y f)_y f = \left( f^T f_{(yy)} + f_y^2 \right) f$

Distribute the multiplication:

$(f_y f)_y f = f^T f_{(yy)} f + f_y^2 f$

Conclusion:

This final expression is exactly what we aimed to prove:

$(f_y f )_y f = f^T f_{(yy)} f + f_y^2 f$

Thus, the identity is shown to be correct.

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Would you like further details on any part of the proof, or do you have any questions? Here are some related questions to explore:

1. How does the Jacobian matrix $f_y$ relate to the derivative of a vector-valued function?
2. What is the significance of the Hessian matrix in this context?
3. Can this identity be extended to functions $f: \mathbb{R}^n \to \mathbb{R}^p$?
4. How does the product rule apply in the context of matrix calculus?
5. What are some applications of such identities in optimization?

Tip: When dealing with matrix derivatives, keeping track of indices and applying the product rule systematically can simplify complex derivations.