بفافي مصفوفة

${\displaystyle A={\begin{bmatrix}0&a\\-a&0\end{bmatrix}}.\qquad \operatorname {pf(A)} =a.}$

${\displaystyle B={\begin{bmatrix}0&a&b\\-a&0&c\\-b&-c&0\end{bmatrix}}.\qquad \operatorname {pf(B)} =0.}$

(3 is odd, so Pfaffian of B is 0)

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

The Pfaffian of a 2n × 2n skew-symmetric tridiagonal matrix is given as

${\displaystyle \operatorname {pf} {\begin{bmatrix}0&a_{1}\\-a_{1}&0&b_{1}\\0&-b_{1}&0&a_{2}\\0&0&-a_{2}&\ddots &\ddots \\&&&\ddots &&b_{n-1}\\&&&&-b_{n-1}&0&a_{n}\\&&&&&-a_{n}&0\end{bmatrix}}=a_{1}a_{2}\cdots a_{n}.}$

which contains the important case of a 2n × 2n skew-symmetric matrix with 2 × 2 blocks on the diagonal:

${\displaystyle \operatorname {pf} {\begin{bmatrix}{\begin{matrix}0&\lambda _{1}\\-\lambda _{1}&0\end{matrix}}&0&\cdots &0\\0&{\begin{matrix}0&\lambda _{2}\\-\lambda _{2}&0\end{matrix}}&&0\\\vdots &&\ddots &\vdots \\0&0&\cdots &{\begin{matrix}0&\lambda _{n}\\-\lambda _{n}&0\end{matrix}}\end{bmatrix}}=\lambda _{1}\lambda _{2}\cdots \lambda _{n}.}$

(Note that any skew-symmetric matrix can be reduced to this form, see Spectral theory of a skew-symmetric matrix)

• محدد (مصفوفات)

• بوابة رياضيات