Second Derivative of Determinant of Matrix?

In summary, if you are given a matrix ##A## that depends on a variable ##x##, the derivative of its determinant is: ##\partial_x A = A \left(A^{-1} \right)_{ji} \partial_x A_{ij}##.
  • #1
brydustin
205
0
Hi all...


I've read on wikipedia (facepalm) that the first derivative of a determinant is

del(det(A))/del(A_ij) = det(A)*(inv(A))_j,i

If we go to find the second derivative (applying power rule), we get:

del^2(A) / (del(A)_pq) (del (A)_ij) = {del(det(A))/del(A_pq)}*(inv(A))_j,i + det(A)*{del(inv(A)_j,i) / del(A_pq)}

I have no clue how to calculate the derivative of the inverse of a matrix with respect to changing the values in the original matrix:
I.E. del(inv(A)_j,i) / del(A_pq)

Also... would be nice if someone could prove the first statement for the first derivative of the determinant.

Thanks!
 
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  • #2
You can use the closed formula ##\det A = \sum_{\sigma \in \operatorname{Sym}(n)} \operatorname{sgn}(\sigma) \prod_{k=1}^n A_{k\sigma(k)}## and calculate.
 
  • #3
I have written about the first derivative of the determinant here

https://www.physicsforums.com/threa...-cofactor-and-determinant.970419/post-6165630
given a matrix ##A## that depends on some variable ##x##: ##A_{ij}=A_{ij}\left(x\right)##, the derivative of its determinant (##A=\mbox{det}\left(A\right)##) is:

##\partial_x A = A \left(A^{-1} \right)_{ji} \partial_x A_{ij}##

if ##x\to A_{sk}## then ##\partial_{x} A_{ij} \to \delta_{si}\delta_{kj}## so:##\partial_{A_{sk}} A = A \left(A^{-1} \right)_{ks} ##
 
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  • #4
brydustin said:
Hi all...I've read on wikipedia (facepalm) that the first derivative of a determinant is

del(det(A))/del(A_ij) = det(A)*(inv(A))_j,i

If we go to find the second derivative (applying power rule), we get:

del^2(A) / (del(A)_pq) (del (A)_ij) = {del(det(A))/del(A_pq)}*(inv(A))_j,i + det(A)*{del(inv(A)_j,i) / del(A_pq)}

I have no clue how to calculate the derivative of the inverse of a matrix with respect to changing the values in the original matrix:
I.E. del(inv(A)_j,i) / del(A_pq)

Also... would be nice if someone could prove the first statement for the first derivative of the determinant.

Thanks!
Can't you use ##Det(AA^{-1})=Id ## to differentiate?
 
  • #5
WWGD said:
Can't you use ##Det(AA^{-1})=Id ## to differentiate?
... and shouldn't the trace be ##D_{Id}\det A\,##?
 
  • #6
WWGD said:
Can't you use ##Det(AA^{-1})=Id ## to differentiate?
Shouldn't Id in the above equation be just 1? ##Det(I) = 1##.
 
  • #7
Mark44 said:
Shouldn't Id in the above equation be just 1? ##Det(I) = 1##.
Ah,yes,Duh myself.
 

1. What is the second derivative of the determinant of a matrix?

The second derivative of the determinant of a matrix is a mathematical concept that involves calculating the rate of change of the first derivative of the determinant. It represents the curvature of the determinant at a specific point.

2. Why is the second derivative of the determinant of a matrix important?

The second derivative of the determinant of a matrix is important because it provides information about the behavior of the determinant at a specific point. It helps in understanding the stability of a system and identifying critical points.

3. How is the second derivative of the determinant of a matrix calculated?

The second derivative of the determinant of a matrix is calculated by taking the derivative of the first derivative of the determinant. This can be done by using the Leibniz formula or by using the properties of determinants.

4. What does the second derivative of the determinant tell us about the matrix?

The second derivative of the determinant provides information about the behavior of the matrix at a specific point. It helps in determining whether the matrix is stable or unstable, and it also helps in identifying critical points such as maximum or minimum values.

5. In what fields of science is the second derivative of the determinant of a matrix used?

The second derivative of the determinant of a matrix is used in various fields of science, including mathematics, physics, engineering, and economics. It has applications in optimization problems, control theory, and stability analysis of systems, among others.

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