Pfaff's determinant

In mathematics , the determinant of an alternating matrix can always be written as the square of a polynomial of the matrix entries. This polynomial is called the Pfaff's determinant of the matrix. The Pfaff determinant is only non-vanishing for alternating matrices. In this case it is a polynomial of degree . ${\displaystyle (2n\times 2n)}$ ${\displaystyle n}$

definition

Be the set of all partitions of in pairs. There are ( double faculty ) such partitions. Each element can be uniquely identified as a ${\displaystyle \Pi }$ ${\displaystyle \{1,2,\ldots ,2n\}}$ ${\displaystyle (2n-1)!!}$ ${\displaystyle \alpha \in \Pi }$

${\displaystyle \alpha =\{(i_{1},j_{1}),(i_{2},j_{2}),\cdots ,(i_{n},j_{n})\}}$

are written with and . Be ${\displaystyle i_{k}<j_{k}}$ ${\displaystyle i_{1}<i_{2}<\ldots <i_{n}}$

${\displaystyle \pi ={\begin{bmatrix}1&2&3&4&\cdots &2n\\i_{1}&j_{1}&i_{2}&j_{2}&\cdots &j_{n}\end{bmatrix}}}$

the corresponding permutation and be the sign of . ${\displaystyle \operatorname {sgn} (\alpha )}$ ${\displaystyle \pi }$

Let be an alternating matrix. For each partition as written above, set ${\displaystyle A=(a_{ij})}$ ${\displaystyle (2n\times 2n)}$ ${\displaystyle \alpha }$

${\displaystyle A_{\alpha }=\operatorname {sgn} (\alpha )a_{i_{1},j_{1}}a_{i_{2},j_{2}}\cdots a_{i_{n},j_{n}}.}$

The Pfaff determinant is then defined as ${\displaystyle A}$

${\displaystyle \operatorname {Pf} (A)=\sum _{\alpha \in \Pi }A_{\alpha }}$.

If odd, then the Pfaff determinant of an alternating matrix is ​​defined as zero. ${\displaystyle m}$ ${\displaystyle (m\times m)}$

Alternative definition

One can each alternate matrix a bivector associate: ${\displaystyle (2n\times 2n)}$ ${\displaystyle A=(a_{ij})}$

${\displaystyle \omega =\sum _{i<j}a_{ij}\;e_{i}\wedge e_{j}}$,

where the default base is for . The Pfaff determinant is defined by ${\displaystyle \{e_{1},e_{2},\ldots ,e_{2n}\}}$ ${\displaystyle \mathbb {R} ^{2n}}$

${\displaystyle {\frac {1}{n!}}\omega ^{n}={\mbox{Pf}}(A)\;e_{1}\wedge e_{2}\wedge \cdots \wedge e_{2n}}$,

here denotes the wedge product of copies of oneself. ${\displaystyle \omega ^{n}}$ ${\displaystyle n}$ ${\displaystyle \omega }$

Examples

${\displaystyle {\mbox{Pf}}{\begin{bmatrix}0&a\\-a&0\end{bmatrix}}=a.}$

${\displaystyle {\mbox{Pf}}{\begin{bmatrix}0&a&b&c\\-a&0&d&e\\-b&-d&0&f\\-c&-e&-f&0\end{bmatrix}}=af-be+dc.}$

${\displaystyle {\mbox{Pf}}{\begin{bmatrix}0&\lambda _{1}&0&0&\cdots &&0\\-\lambda _{1}&0&\omega _{1}&0&&&\\0&-\omega _{1}&0&\lambda _{2}&&&\vdots \\0&0&-\lambda _{2}&\ddots &\ddots &&\\\vdots &&&\ddots &\ddots &\omega _{n-1}&\\&&&&-\omega _{n-1}&0&\lambda _{n}\\0&&\cdots &&&-\lambda _{n}&0\end{bmatrix}}=\lambda _{1}\lambda _{2}\cdots \lambda _{n}.}$

properties

For an alternating -Matrix and any -Matrix holds ${\displaystyle (2n\times 2n)}$ ${\displaystyle A}$ ${\displaystyle (2n\times 2n)}$ ${\displaystyle B}$

• ${\displaystyle {\mbox{Pf}}(A)^{2}=\det(A)}$

• ${\displaystyle {\mbox{Pf}}(BAB^{T})=\det(B){\mbox{Pf}}(A)}$

• ${\displaystyle {\mbox{Pf}}(\lambda A)=\lambda ^{n}{\mbox{Pf}}(A)}$

• ${\displaystyle {\mbox{Pf}}(A^{T})=(-1)^{n}{\mbox{Pf}}(A)}$

• For a block diagonal matrix

${\displaystyle A_{1}\oplus A_{2}={\begin{bmatrix}A_{1}&0\\0&A_{2}\end{bmatrix}}}$

applies . ${\displaystyle {\mbox{Pf}}(A_{1}\oplus A_{2})={\mbox{Pf}}(A_{1}){\mbox{Pf}}(A_{2})}$

• For any matrix : ${\displaystyle (n\times n)}$ ${\displaystyle M}$

${\displaystyle {\mbox{Pf}}{\begin{bmatrix}0&M\\-M^{T}&0\end{bmatrix}}=(-1)^{n(n-1)/2}\det M.}$

Applications

The Pfaff determinant is an invariant polynomial of an alternating matrix (note: it is not invariant under general base changes, but only under orthogonal transformations). As such, it is important for the theory of characteristic classes . (In this context it is also called the Euler polynomial .) In particular, it can be used to define the Euler class of a Riemannian manifold . This is used in the Gauss-Bonnet theorem .

The number of perfect pairings in a planar graph is equal to the absolute value of a suitable Pfaff determinant, which can be calculated in polynomial time. This is particularly surprising because the problem is very difficult for general graphs ( Sharp-P -complete). The result is used in physics to calculate the sum of properties of the Ising model of spin glasses . The underlying graph is planar. It has also recently been used to develop efficient algorithms for otherwise seemingly unsolvable problems. This includes the efficient simulation of certain types of quantum calculations.

history

The term Pfaff's determinant was coined by Arthur Cayley , who used it in 1852: "The permutants of this class (from their connection with the researches of Pfaff on differential equations) I shall term Pfaffians ." This was done in honor of the German mathematician Johann Friedrich Pfaff .

See also

• Invariant polynomial

Web links

• Pfaffian at PlanetMath.org (English)