I am trying to solve for the derivative of the following equation:

$t(\pmb {m_1} , \pmb {m_2} )= \frac{1}{2} \: ||\nabla \pmb {m_1} ||_2^2 \: ||\nabla \pmb {m_2}||_2^2 \: - \frac{1}{2} \:|\nabla \pmb {m_1} \cdot \nabla \pmb {m_2} |^2 $

where $\pmb {m_1}$ and $\pmb {m_2}$ are matrices that have been vectorized, and $\nabla \pmb {m_1}$, $\nabla \pmb {m_2}$ are the gradients that have been vectorized as well. Gradients are calculated using finite difference along x, y and z dimensions.

Basically, I am trying to solve for $ \frac {\partial t}{\partial \pmb {m_1}} $ and $ \frac {\partial t}{\partial \pmb {m_2}} $.

So far I have this:

$ \frac {\partial t}{\partial \pmb {m_1}} = (\nabla \pmb {m_1} \cdot \frac {\partial \nabla \pmb {m_1}}{\partial \pmb {m_1}}) \: ||\nabla \pmb {m_2}||_2^2 \: - (\nabla \pmb {m_1} \cdot \nabla \pmb {m_2})\cdot(\frac {\partial \nabla \pmb {m_1}}{\partial \pmb {m_1}} \cdot \nabla \pmb {m_2})$

$ \frac {\partial t}{\partial \pmb {m_2}} = (\nabla \pmb {m_2} \cdot \frac {\partial \nabla \pmb {m_2}}{\partial \pmb {m_2}}) \: ||\nabla \pmb {m_1}||_2^2 \: - (\nabla \pmb {m_1} \cdot \nabla \pmb {m_2})\cdot(\frac {\partial \nabla \pmb {m_2}}{\partial \pmb {m_2}} \cdot \nabla \pmb {m_1})$

I have trouble understanding $\frac {\partial \nabla \pmb {m_1}}{\partial \pmb {m_1}}$ and $\frac {\partial \nabla \pmb {m_2}}{\partial \pmb {m_2}}$

My intuition tells me these just equal to $I$ (the identity matrix), but I am not sure how to prove it.

This is how I have proceeded so far.

$\frac {\partial \nabla \pmb m}{\partial \pmb m} = \left [\begin{matrix} \frac {\partial \nabla_1}{\partial m_1} & \frac {\partial \nabla_1}{\partial m_2} & \cdots & \frac {\partial \nabla_1}{\partial m_M} \\ \frac {\partial \nabla_2}{\partial m_1} & \frac {\partial \nabla_2}{\partial m_1} & \cdots & \frac {\partial \nabla_2}{\partial m_M} \\ \vdots & \vdots & \ddots & \vdots \\ \frac {\partial \nabla_M}{\partial m_1} & \frac {\partial \nabla_M}{\partial m_2} & \cdots & \frac {\partial \nabla_M}{\partial m_M} \end{matrix} \right ]$


$\nabla_1 = (\frac {\partial m_1}{\partial x}, \frac {\partial m_1}{\partial y}, \frac {\partial m_1}{\partial z})$

$\nabla_2 = (\frac {\partial m_2}{\partial x}, \frac {\partial m_2}{\partial y}, \frac {\partial m_2}{\partial z})$



$\nabla_M = (\frac {\partial m_M}{\partial x}, \frac {\partial m_M}{\partial y}, \frac {\partial m_M}{\partial z})$

which leads me to think that it is equal to 1 along the diagonal and 0 everywhere else.

Anyone who has experience with these problems, could you please help me? Otherwise, it would be deeply appreciated if you could refer me to any references where I can look deeper into this.

Thank you.


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