A SPECIAL Derivative of Matrix Determinant (tensor involved)

In summary, the conversation discusses a surface defined by an implicit function, with a gradient and Hessian matrix calculated for the surface. The goal is to find the derivative of the determinant of the Hessian matrix with respect to a point on the surface. The formula for calculating this derivative is found, but there is a problem with the resulting 3x3x3 tensor. The person is asking for help in finding the correct rule to use in this situation to obtain a 3x1 vector instead.
  • #1
roger1318
3
0
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" ), 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!
 
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  • #2
Trying to avoid bulk work by fancy notation? Or you took the partial derivatives by hand before posting...
 

FAQ: A SPECIAL Derivative of Matrix Determinant (tensor involved)

What is a special derivative of matrix determinant?

A special derivative of matrix determinant is a way to calculate the rate of change of a matrix determinant with respect to a variable, such as time or another matrix element. It is an important tool in the study of linear transformations and vector calculus.

How is a special derivative of matrix determinant calculated?

To calculate a special derivative of matrix determinant, you must first expand the determinant using cofactor expansion. Then, you can differentiate each term using standard calculus rules. Finally, you can evaluate the resulting derivatives at a specific value of the variable in question.

What is the application of a special derivative of matrix determinant?

A special derivative of matrix determinant has many applications in fields such as physics, engineering, and computer graphics. It can be used to study the stability of systems, optimize functions, and model physical phenomena such as fluid flow and electric fields.

Is a special derivative of matrix determinant different from a regular derivative?

Yes, a special derivative of matrix determinant is different from a regular derivative in that it involves matrices and tensors instead of just scalar values. It also follows different rules and properties due to the unique nature of matrices and determinants.

Are there any special properties or theorems related to a special derivative of matrix determinant?

Yes, there are several special properties and theorems related to a special derivative of matrix determinant, such as the chain rule, product rule, and quotient rule. There are also several identities and properties specific to matrix determinants, such as the Leibniz formula and the Jacobi identity.

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