There is a surface defined by setting implicit function g((adsbygoogle = window.adsbygoogle || []).push({}); x)=0, wherexis a 3 by 1 column vector, denoting a point on the surface;

3X1 vector [tex]\nabla[/tex]g(x) is the Gradient(surface normal at pointx;

3X3 matrixH(g(x)) = [tex]\nabla^2[/tex](g(x)) is the Hessian Matrix;

3X3X3 tensor [tex]\nabla^3g(\bold{x})[/tex] is [tex]\frac{\partial \bold{H}}{\partial \bold{x}}[/tex]

The goal is to find [tex]{\color{red}\frac{\partial (det(\bold{H}))}{\partial \bold{x}}}[/tex] ; which should be a 3X1 vector since the determinant ofHis a scalar.

I found the formula for calculating the derivative of the determinant of a square matrix with respect to itself at (http://en.wikipedia.org/wiki/Matrix_calculus" [Broken]), which in my case here is a 3X3 matrix.

[tex]\frac{\partial (det(\bold{H}))}{\partial \bold{H}}=det(\bold{H})\cdot\bold{H}^-^1}[/tex]

But what I want is the derivative with respect to that pointx, not with respect to the matrix itself in the conventional sense.

So I attempted to use chain rule as we do in most cases:

[tex]\frac{\partial (det(\bold{H}))}{\partial \bold{x}}=\frac{\partial (det(\bold{H}))}{\partial \bold{H}}\cdot\frac{\partial \bold{H}}{\partial \bold{x}}=det(\bold{H})\cdot \underline{\bold{H}^-^1}}\cdot \underline{\underline{\nabla^3 g(\bold{x})}} [/tex]

Now here comes the problem:

[tex]{\color{blue}\bold{H}^-^1}}[/tex] is a 3X3 matrix and [tex]{\color{blue}\nabla^3g(\bold{x})}[/tex] is 3X3X3 tensor; their multiplication product is still a 3X3X3 tensor but not the 3X1 vector as expected.

I am pretty sure something must've gone wrong; anybody could tell me where? And how am I supposed to deal with this determinant derivative issue? If for some reason the chain rule doesn't apply here, what rule should I use to get the 3X1 vector?

Any comments are much appreciated!!!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A SPECIAL Derivative of Matrix Determinant (tensor involved)

**Physics Forums | Science Articles, Homework Help, Discussion**