Mixing
Derivative of gradient of vector wrt vector
$begingroup$
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.
$frac {partial nabla pmb m}{partial pmb m} = left [begin{matrix} frac {partial nabla_1}{partial m_1} & frac {partial nabla_1}{partial m_2} & cdots & frac {partial nabla_1}{partial m_M} \ frac {partial nabla_2}{partial m_1} & frac {partial nabla_2}{partial m_1} & cdots & frac {partial nabla_2}{partial m_M} \ vdots & vdots & ddots & vdots \ frac {partial nabla_M}{partial m_1} & frac {partial nabla_M}{partial m_2} & cdots & frac {partial nabla_M}{partial m_M}
end{matrix} right ]$
where,
$nabla_1 = (frac {partial m_1}{partial x}, frac {partial m_1}{partial y}, frac {partial m_1}{partial z})$
$nabla_2 = (frac {partial m_2}{partial x}, frac {partial m_2}{partial y}, frac {partial m_2}{partial z})$
...
...
$nabla_M = (frac {partial m_M}{partial x}, frac {partial m_M}{partial y}, frac {partial m_M}{partial z})$
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.
calculus matrices derivatives vectors vector-analysis
asked Jan 11 at 0:24
user633611user633611
11
$endgroup$
add a comment |
$begingroup$
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.
$frac {partial nabla pmb m}{partial pmb m} = left [begin{matrix} frac {partial nabla_1}{partial m_1} & frac {partial nabla_1}{partial m_2} & cdots & frac {partial nabla_1}{partial m_M} \ frac {partial nabla_2}{partial m_1} & frac {partial nabla_2}{partial m_1} & cdots & frac {partial nabla_2}{partial m_M} \ vdots & vdots & ddots & vdots \ frac {partial nabla_M}{partial m_1} & frac {partial nabla_M}{partial m_2} & cdots & frac {partial nabla_M}{partial m_M}
end{matrix} right ]$
where,
$nabla_1 = (frac {partial m_1}{partial x}, frac {partial m_1}{partial y}, frac {partial m_1}{partial z})$
$nabla_2 = (frac {partial m_2}{partial x}, frac {partial m_2}{partial y}, frac {partial m_2}{partial z})$
...
...
$nabla_M = (frac {partial m_M}{partial x}, frac {partial m_M}{partial y}, frac {partial m_M}{partial z})$
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.
calculus matrices derivatives vectors vector-analysis
asked Jan 11 at 0:24
user633611user633611
11
$endgroup$
add a comment |
$begingroup$
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.
$frac {partial nabla pmb m}{partial pmb m} = left [begin{matrix} frac {partial nabla_1}{partial m_1} & frac {partial nabla_1}{partial m_2} & cdots & frac {partial nabla_1}{partial m_M} \ frac {partial nabla_2}{partial m_1} & frac {partial nabla_2}{partial m_1} & cdots & frac {partial nabla_2}{partial m_M} \ vdots & vdots & ddots & vdots \ frac {partial nabla_M}{partial m_1} & frac {partial nabla_M}{partial m_2} & cdots & frac {partial nabla_M}{partial m_M}
end{matrix} right ]$
where,
$nabla_1 = (frac {partial m_1}{partial x}, frac {partial m_1}{partial y}, frac {partial m_1}{partial z})$
$nabla_2 = (frac {partial m_2}{partial x}, frac {partial m_2}{partial y}, frac {partial m_2}{partial z})$
...
...
$nabla_M = (frac {partial m_M}{partial x}, frac {partial m_M}{partial y}, frac {partial m_M}{partial z})$
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.
calculus matrices derivatives vectors vector-analysis
asked Jan 11 at 0:24
user633611user633611
11
$endgroup$
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.
$frac {partial nabla pmb m}{partial pmb m} = left [begin{matrix} frac {partial nabla_1}{partial m_1} & frac {partial nabla_1}{partial m_2} & cdots & frac {partial nabla_1}{partial m_M} \ frac {partial nabla_2}{partial m_1} & frac {partial nabla_2}{partial m_1} & cdots & frac {partial nabla_2}{partial m_M} \ vdots & vdots & ddots & vdots \ frac {partial nabla_M}{partial m_1} & frac {partial nabla_M}{partial m_2} & cdots & frac {partial nabla_M}{partial m_M}
end{matrix} right ]$
where,
$nabla_1 = (frac {partial m_1}{partial x}, frac {partial m_1}{partial y}, frac {partial m_1}{partial z})$
$nabla_2 = (frac {partial m_2}{partial x}, frac {partial m_2}{partial y}, frac {partial m_2}{partial z})$
...
...
$nabla_M = (frac {partial m_M}{partial x}, frac {partial m_M}{partial y}, frac {partial m_M}{partial z})$
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.
calculus matrices derivatives vectors vector-analysis
calculus matrices derivatives vectors vector-analysis
asked Jan 11 at 0:24
user633611user633611
11
asked Jan 11 at 0:24
user633611user633611
11
asked Jan 11 at 0:24
user633611user633611
11
asked Jan 11 at 0:24
user633611user633611
11
asked Jan 11 at 0:24
user633611user633611
11
11
add a comment |
add a comment |
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