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A SPECIAL Derivative of Matrix Determinant (tensor involved)

  1. Oct 13, 2009 #1
    There is a surface defined by setting implicit function g(x)=0, where x is 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 point x;

    3X3 matrix H(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 of H is 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 point x, 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!!!
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Oct 13, 2009 #2
    Trying to avoid bulk work by fancy notation? Or you took the partial derivatives by hand before posting...
     
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