Determinant of Matrix Component

• member 428835
I'll try it now.In summary, the partial derivative of the determinant of a matrix A with respect to any element A_{pq} is given by (1/2) * epsilon_{pjk} * epsilon_{qmn} * A_{jm} * A_{kn}. This can be obtained by permuting the rows and columns of the determinant and using the properties of the Levi-Civita symbol.
member 428835

Homework Statement

Show $$\frac{\partial \det(A)}{\partial A_{pq}} = \frac{1}{2}\epsilon_{pjk}\epsilon_{qmn}A_{jm}A_{kn}$$

Homework Equations

##\det(A)=\epsilon_{ijk}A_{1i}A_{2j}A_{3k}##

The Attempt at a Solution

$$\frac{\partial \det(A)}{\partial A_{pq}}=\frac{\partial}{\partial A_{pq}}\epsilon_{ijk}A_{1i}A_{2j}A_{3k}\\ =\epsilon_{ijk}\frac{\partial}{\partial A_{pq}}A_{1i}A_{2j}A_{3k}$$
but I'm now stuck. I feel like one of the ##A## components on the RHS must go to 1 and the rest would be constant, leaving some sort of ##\epsilon A_{yy} A_{xx}## behind. I'm thinking along the lines of ##\partial A_{ij}/\partial A_{ik} = \delta_{jk}##. Any ideas?

joshmccraney said:
Any ideas?

I'd be tempted to move ##A_{pq}## to the top-left of the matrix.

PeroK said:
I'd be tempted to move ##A_{pq}## to the top-left of the matrix.
Are you suggesting take
$$\frac{\partial \det(A)}{\partial A_{11}}=\frac{\partial}{\partial A_{11}}\epsilon_{ijk}A_{1i}A_{2j}A_{3k}\\ =\epsilon_{ijk}A_{2j}A_{3k}\frac{\partial}{\partial A_{11}}A_{1i}\\ =\epsilon_{1jk}A_{2j}A_{3k}$$

joshmccraney said:
Are you suggesting take
$$\frac{\partial \det(A)}{\partial A_{11}}=\frac{\partial}{\partial A_{11}}\epsilon_{ijk}A_{1i}A_{2j}A_{3k}\\ =\epsilon_{ijk}A_{2j}A_{3k}\frac{\partial}{\partial A_{11}}A_{1i}\\ =\epsilon_{1jk}A_{2j}A_{3k}$$

One way to do it is just to do it for all 9 elements. Not very elegant.

Otherwise, you need to get a general expression for the terms that have ##A_{pq}## in them. It might be easier to rearrange the matrix, getting ##A_{pq}## to the top left corner.

PeroK said:
One way to do it is just to do it for all 9 elements. Not very elegant.
Agreed!

PeroK said:
Otherwise, you need to get a general expression for the terms that have ##A_{pq}## in them. It might be easier to rearrange the matrix, getting ##A_{pq}## to the top left corner.
What do you mean "general expression"? Also, you suggest getting ##A_{pq}## to the top left, doesn't this mean making ##A_{pq}=A_{11}##?

joshmccraney said:
Agreed!

What do you mean "general expression"? Also, you suggest getting ##A_{pq}## to the top left, doesn't this mean making ##A_{pq}=A_{11}##?

It means using row and column operations. Alternatively, express the determinant in an equivalent form for expansion via the ##p##th row.

PeroK said:
It means using row and column operations. Alternatively, express the determinant in an equivalent form for expansion via the ##p##th row.
Since the determinant is a scalar, I'm unsure how to express it for the ##p##th row. Could you elaborate?

joshmccraney said:
Since the determinant is a scalar, I'm unsure how to express it for the ##p##th row. Could you elaborate?

You can evaluate a determinant using any row as your starting row.

But, in fact, if you permute the rows to get the ##p##th row at the top, with row ##p+1## second and row ##p+2## third (where these are numbers modulo 3), I think it comes out easily enough. Then permute the columns to get the colums in order ##q, q+1, q+2##.

Then, if you do the same trick with the Levi-Civita, I think the equation should drop out for general ##p, q##.

PS by the same trick, I mean that:

##\epsilon_{pjk} = 1## when ##j = p+1## and ##k = p+2##
##\epsilon_{pjk} = -1## when ##j = p+2## and ##k = p+1##

And is ##0## otherwise.

PeroK said:
You can evaluate a determinant using any row as your starting row.
I agree.

PeroK said:
But, in fact, if you permute the rows to get the ##p##th row at the top, with row ##p+1## second and row ##p+2## third (where these are numbers modulo 3), I think it comes out easily enough. Then permute the columns to get the colums in order ##q, q+1, q+2##.
I don't think I'm understanding what you're saying. Could you illustrate, perhaps with a different problem? I just don't know the technique you suggest.

PeroK said:
PS by the same trick, I mean that:

##\epsilon_{pjk} = 1## when ##j = p+1## and ##k = p+2##
##\epsilon_{pjk} = -1## when ##j = p+2## and ##k = p+1##

And is ##0## otherwise.
I agree.

Suppose we take the case ##p, q = 1, 1##:

##det(A) = A_{11}(A_{22}A_{33} - A_{23}A_{32}) + \dots##

Where the rest of the terms do not involve ##A_{11}##, hence you can calculate the required partial derivative.

You can also check fairly easily that:

##\frac{1}{2}\epsilon_{1jk}\epsilon_{1mn}A_{jm}A_{kn} = A_{22}A_{33} - A_{23}A_{32}##

Now, by either two row operations or no row operations and two column operations or no column operations, you permute ##A_{pq}## to the top left of your determinant. You need to check this for ##p, q = 1, 2, 3##. And, this gives

##det(A) = A_{pq}(A_{p+1, q+1}A_{p+2, q+2} - A_{p+1, q+2}A_{p+2, q+1}) + \dots##

Where we are using modulo 3 here. And that's just about it. All you need to do is manipulate the expression involving the Levi_Civita coefficients using a similar modulo 3 approach.

member 428835
Thanks!

1. What is the determinant of a matrix component?

The determinant of a matrix component is a value that represents the scaling factor of the matrix. It is calculated using specific mathematical operations on the elements of the matrix.

2. Why is the determinant important?

The determinant is important because it provides valuable information about the properties and behavior of a matrix. It can determine if a matrix is invertible, if it has linearly independent rows and columns, and can be used to solve systems of linear equations.

3. How is the determinant calculated?

The determinant of a matrix is calculated by expanding along a row or column and multiplying each element by its corresponding minor (the determinant of the submatrix formed by removing that element) and then alternating the signs of the products. These products are then summed to obtain the determinant value.

4. Are there any properties of the determinant?

Yes, there are several properties of the determinant including the fact that it is only defined for square matrices, it is equal to zero if the matrix is singular, and it follows the rule of multiplication by a constant and addition of matrices.

5. How is the determinant used in real life applications?

The determinant is used in a variety of real life applications such as solving systems of linear equations in engineering and physics, calculating areas and volumes in geometry, and in computer graphics and machine learning algorithms.

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