Adjugate of singular skew-symmetric matrix

[gerald V]

<Moderator's note: Moved from a mathematical forum.>

When one differentiates the determinant of a matrix, the adjugate of the matrix comes into play. The formula holds irrespective of whether or not the determinant vanishes.

I tried this for the 4x4 electromagnetic field tensor $F$. But actually the result would hold for any skew-symmetric 4x4 matrix, since one can regard the components of $E$ and $B$ just as symbols for any numbers, I think.

As is known, the determinant of the electromagnetic field is $(E \cdot B)^2$ up to a numerical factor. When I calculated the adjugate, it turned out that the resulting (again skew-symmetric) matrix is proportional to $E \cdot B$. This means that if the determinant of the electromagnetic field is zero, so is the entire adjugate matrix. However, the variation of $\sqrt{\det F}$ would be sensible even if the determinant is zero - very remarkable.

Is this correct or did I miscalculate?

Thank you very much in Advance.

 

[eljose79]

Hello there,

Thank you for your post. It is always interesting to see applications of mathematical concepts in different fields.

First of all, your understanding of the adjugate matrix is correct. The adjugate matrix is defined as the transpose of the cofactor matrix, which is obtained by replacing each element of the original matrix with its corresponding minor (determinant of the submatrix obtained by deleting the row and column of that element). So, the adjugate matrix is indeed proportional to the original matrix if and only if the determinant of the original matrix is zero.

In the case of the electromagnetic field tensor, the determinant is indeed proportional to $(E \cdot B)^2$, as you mentioned. So, when the determinant is zero, the adjugate matrix will also be zero. This is not surprising, as the two matrices are related through a simple proportionality factor.

However, as you correctly pointed out, the variation of $\sqrt{\det F}$ would still be sensible even when the determinant is zero. This is because the variation takes into account not just the value of the determinant, but also the variation of the elements of the matrix. So, even if the determinant is zero, the variation can still be non-zero if there is a change in the elements of the matrix.

In conclusion, your calculations are correct and your understanding of the adjugate matrix is also correct. Keep exploring and applying mathematical concepts in your field of study. It is always exciting to see cross-disciplinary applications of mathematics.

Best of luck in your research.