普法夫值

若A是2n x 2n的反对称矩阵

${\displaystyle \operatorname {pf} (A^{\text{T}})=(-1)^{n}\operatorname {pf} (A).}$

${\displaystyle \operatorname {pf} (\lambda A)=\lambda ^{n}\operatorname {pf} (A).}$

${\displaystyle \operatorname {pf} (A)^{2}=\det(A).}$

若B是任何2n x 2n矩阵

${\displaystyle \operatorname {pf} (BAB^{\text{T}})=\det(B)\operatorname {pf} (A).}$

若A^TB是正定矩阵

${\displaystyle {\textrm {pf}}(A)\,{\textrm {pf}}(B)=\exp({\tfrac {1}{2}}\mathrm {tr} \log(A^{\text{T}}B)).}$

不然的是

${\displaystyle \mathrm {pf} (A)\,\mathrm {pf} (B)={\tfrac {1}{n!}}B_{n}(s_{1},s_{2},\ldots ,s_{n}),\qquad \mathrm {where} \qquad s_{l}=-{\tfrac {1}{2}}(l-1)!\,\mathrm {tr} ((AB)^{l})}$

其中的B_n(s₁,s₂,...,s_n)是贝尔多项式。

• 陈-高斯-博内定理，欧拉示性数是曲率形式的普法夫值
• 双体模型 [1]

• Zhang, Fuzhen, ed. The Schur complement and its applications. Vol. 4. Springer Science & Business Media, 2006.
• https://web.archive.org/web/20160305005223/http://jesusmtz.public.iastate.edu/soliton/REPORT%202.pdf（孤子）
• http://alexandria.tue.nl/repository/freearticles/597510.pdf

1. Propp, James. . arXiv:math/0406301. 2004-06-15.