Independent indeterminate elements

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I'm trying to figure out what 'skew symmetric matrix in which the elements above the diagonal are independent indeterminates over the ring of rational integers' means.
I'm kind of confused by this wording. I know that a skew-symmetric matrix is defined as $A^T = -A$. The phrase is taken from a text about Pfaffians, determinants of skew-symmetric matrices, and their use for perfect matching of graphs using FKT algorithm.



If I'm not mistaken



$
A = beginpmatrix
0 & a & b & c \
-a & 0 & d & e \
-b & -d &0 & f \
-c & -e & -f & 0
endpmatrix
$



is an example of a skew-symmetric matrix. The elements above the diagonal are $a, b, c, d, e, f$, and their negated values are below the diagonal. So, what does 'independent indeterminate' means for these elements?




matrices definition pfaffian




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  • I would speculate it means that $a,b,c,d,e,f$ is an exhaustive list of the independent variables which take values from the ring of rational integers and presumably because they will want to show the determinant to be a polynomial in these variables.
    – CyclotomicField
    Jul 30 at 17:49














up vote
0
down vote

favorite












I'm trying to figure out what 'skew symmetric matrix in which the elements above the diagonal are independent indeterminates over the ring of rational integers' means.
I'm kind of confused by this wording. I know that a skew-symmetric matrix is defined as $A^T = -A$. The phrase is taken from a text about Pfaffians, determinants of skew-symmetric matrices, and their use for perfect matching of graphs using FKT algorithm.



If I'm not mistaken



$
A = beginpmatrix
0 & a & b & c \
-a & 0 & d & e \
-b & -d &0 & f \
-c & -e & -f & 0
endpmatrix
$



is an example of a skew-symmetric matrix. The elements above the diagonal are $a, b, c, d, e, f$, and their negated values are below the diagonal. So, what does 'independent indeterminate' means for these elements?




matrices definition pfaffian




share|cite|improve this question



















  • I would speculate it means that $a,b,c,d,e,f$ is an exhaustive list of the independent variables which take values from the ring of rational integers and presumably because they will want to show the determinant to be a polynomial in these variables.
    – CyclotomicField
    Jul 30 at 17:49












up vote
0
down vote

favorite









up vote
0
down vote

favorite











I'm trying to figure out what 'skew symmetric matrix in which the elements above the diagonal are independent indeterminates over the ring of rational integers' means.
I'm kind of confused by this wording. I know that a skew-symmetric matrix is defined as $A^T = -A$. The phrase is taken from a text about Pfaffians, determinants of skew-symmetric matrices, and their use for perfect matching of graphs using FKT algorithm.



If I'm not mistaken



$
A = beginpmatrix
0 & a & b & c \
-a & 0 & d & e \
-b & -d &0 & f \
-c & -e & -f & 0
endpmatrix
$



is an example of a skew-symmetric matrix. The elements above the diagonal are $a, b, c, d, e, f$, and their negated values are below the diagonal. So, what does 'independent indeterminate' means for these elements?




matrices definition pfaffian




share|cite|improve this question











I'm trying to figure out what 'skew symmetric matrix in which the elements above the diagonal are independent indeterminates over the ring of rational integers' means.
I'm kind of confused by this wording. I know that a skew-symmetric matrix is defined as $A^T = -A$. The phrase is taken from a text about Pfaffians, determinants of skew-symmetric matrices, and their use for perfect matching of graphs using FKT algorithm.



If I'm not mistaken



$
A = beginpmatrix
0 & a & b & c \
-a & 0 & d & e \
-b & -d &0 & f \
-c & -e & -f & 0
endpmatrix
$



is an example of a skew-symmetric matrix. The elements above the diagonal are $a, b, c, d, e, f$, and their negated values are below the diagonal. So, what does 'independent indeterminate' means for these elements?





matrices definition pfaffian





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 30 at 17:43









kurapan

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  • I would speculate it means that $a,b,c,d,e,f$ is an exhaustive list of the independent variables which take values from the ring of rational integers and presumably because they will want to show the determinant to be a polynomial in these variables.
    – CyclotomicField
    Jul 30 at 17:49
















  • I would speculate it means that $a,b,c,d,e,f$ is an exhaustive list of the independent variables which take values from the ring of rational integers and presumably because they will want to show the determinant to be a polynomial in these variables.
    – CyclotomicField
    Jul 30 at 17:49















I would speculate it means that $a,b,c,d,e,f$ is an exhaustive list of the independent variables which take values from the ring of rational integers and presumably because they will want to show the determinant to be a polynomial in these variables.
– CyclotomicField
Jul 30 at 17:49




I would speculate it means that $a,b,c,d,e,f$ is an exhaustive list of the independent variables which take values from the ring of rational integers and presumably because they will want to show the determinant to be a polynomial in these variables.
– CyclotomicField
Jul 30 at 17:49















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