Pfaffian and determinants of skew symmetric matrices

In summary, the conversation is about understanding operators and determinants for skew symmetric matrices, specifically the Moore determinant and its polarization of mixed determinants. The person is looking for resources or someone to explain the construction process and restrictions on the base vector space. They also mention trying to use the operators on n-dimensional split quaternions.
  • #1
camilus
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Can anyone explain or point me to a good resource to understand these operators? I'm trying to the understand determinants for skew symmetric matrices, more specifically the Moore determinant and it's polarization of mixed determinants. Can hone shed some light? I'm confused as to how the construction is done in the first place and what are the restrictions on the base Vector space.

I'm trying to see if I can use the on the n-dimensional split quaternions.
Thanks!
CM
 
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  • #2
Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?
 

1. What is a Pfaffian of a skew symmetric matrix?

A Pfaffian is a function that is defined for skew symmetric matrices. It is similar to a determinant, but instead of the product of diagonal elements, it is the sum of all possible products of non-diagonal elements. It is denoted by Pf(A) or Pfaff(A).

2. How is the Pfaffian calculated?

The Pfaffian can be calculated using the following formula: Pf(A) = 1/2^n * sum over all permutations of (a_ij * a_kl * ... * a_nm), where n is the size of the matrix and a_ij represents the element in the i-th row and j-th column. Alternatively, there are also specific algorithms and software programs that can calculate the Pfaffian efficiently.

3. What is the significance of the Pfaffian in mathematics?

The Pfaffian has many applications in mathematics, especially in areas such as combinatorics, graph theory, and theoretical physics. It is also closely related to other important concepts such as alternating forms, exterior algebra, and Grassmann algebra.

4. Can the Pfaffian of a skew symmetric matrix be negative?

No, the Pfaffian of a skew symmetric matrix is always non-negative. This is because the Pfaffian is defined as a sum of products, and the product of two negative numbers is always positive.

5. How is the Pfaffian related to the determinant of a skew symmetric matrix?

The Pfaffian is closely related to the determinant of a skew symmetric matrix. In fact, for a skew symmetric matrix of odd size, the Pfaffian is equal to the square root of the determinant. For a skew symmetric matrix of even size, the Pfaffian is equal to the square root of the absolute value of the determinant. Additionally, the Pfaffian and determinant have similar properties and can be used to solve similar problems.

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