Pfaffian

A Pfaffian of a skew - symmetric matrix is a polynomial in its elements whose square is equal to the determinant of this matrix. Like the determinant, Pfaffian is nonzero only for skew-symmetric matrices of size $2n\times 2n$ ${\ displaystyle 2n \ times 2n}$ $2n \ times 2n$ , and in this case its degree is n .

Examples

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

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

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

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

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

Definition

Let be $\Pi$ ${\ displaystyle \ Pi}$ denotes the set of all partitions of the set $\{1,2,\dots ,2n\}$ ${\ displaystyle \ {1,2, \ dots, 2n \}}$ unordered pairs (total exists $(2n-1)!!$ ${\ displaystyle (2n-1) !!}$ such partitions). Breaking up $\alpha \in \Pi$ ${\ displaystyle \ alpha \ in \ Pi}$ can be recorded

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

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

Where $i_{k}<j_{k}$ ${\ displaystyle i_ {k} <j_ {k}}$ and $i_{1}<i_{2}<\cdots <i_{n}$ ${\ displaystyle i_ {1} <i_ {2} <\ cdots <i_ {n}}$ . Let be

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

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

denotes the corresponding permutation , and ${\mbox{sgn}}(\alpha )$ ${\ displaystyle {\ mbox {sgn}} (\ alpha)}$ - permutation sign $\pi$ ${\ displaystyle \ pi}$ . It is easy to see that ${\mbox{sgn}}(\alpha )$ ${\ displaystyle {\ mbox {sgn}} (\ alpha)}$ independent of choice $\pi$ ${\ displaystyle \ pi}$ .

Let be $A=\{a_{ij}\}$ ${\ displaystyle A = \ {a_ {ij} \}}$ denotes $2n\times 2n$ {\ displaystyle 2n \ times 2n} skew-symmetric matrix. For splitting $\alpha$ ${\ displaystyle \ alpha}$ define

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

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

Now we can define the Pfaffian of the matrix A as

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

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

Pfaffian skew-symmetric matrix size $n\times n$ ${\ displaystyle n \ times n}$ for odd n is equal to zero by definition.

Recursive Definition

Pfaffian size matrix $0\times 0$ ${\ displaystyle 0 \ times 0}$ set equal to 1; Pfaffian of a skew-symmetric matrix A of size $2n\times 2n$ ${\ displaystyle 2n \ times 2n}$ at $n>0$ ${\ displaystyle n> 0}$ can be defined recursively as follows:

$\operatorname {Pf} (A)=\sum _{{j=1} \atop {j\neq i}}^{2n}(-1)^{i+j+1+\theta (i-j)}a_{ij}\operatorname {Pf} (A_{{\hat {\imath }}{\hat {\jmath }}}),$ ${\ displaystyle \ operatorname {Pf} (A) = \ sum _ {{j = 1} \ atop {j \ neq i}} ^ {2n} (- 1) ^ {i + j + 1 + \ theta (ij )} a_ {ij} \ operatorname {Pf} (A _ {{\ hat {\ imath}} {\ hat {\ jmath}}},}$

where is the index $i$ ${\ displaystyle i}$ can be chosen arbitrarily $\theta (i-j)$ ${\ displaystyle \ theta (ij)}$ - Heaviside function , $A_{{\hat {\imath }}{\hat {\jmath }}}$ ${\ displaystyle A _ {{\ hat {\ imath}} {\ hat {\ jmath}}}}$ denotes a matrix A without the ith and jth columns and rows.

Alternative Definition

For $2n\times 2n$ ${\ displaystyle 2n \ times 2n}$ skew-symmetric matrix $A=\{a_{ij}\}$ ${\ displaystyle A = \ {a_ {ij} \}}$ consider the bivector :

\omega =\sum _{i<j}a_{ij}\;e_{i}\wedge e_{j}.

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

Where $\{e_{1},e_{2},\dots ,e_{2n}\}$ ${\ displaystyle \ {e_ {1}, e_ {2}, \ dots, e_ {2n} \}}$ there is a standard basis in $\mathbb {R} ^{2n}$ ${\ displaystyle \ mathbb {R} ^ {2n}}$ . Then Pfaffian is defined by the following equation:

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

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

Where $\omega ^{\wedge n}$ ${\ displaystyle \ omega ^ {\ wedge n}}$ denotes the outer product of n copies $\omega$ ${\ displaystyle \ omega}$ .

Properties

For $2n\times 2n$ ${\ displaystyle 2n \ times 2n}$ skew-symmetric matrix $A$ ${\ displaystyle A}$ and for arbitrary $2n\times 2n$ ${\ displaystyle 2n \ times 2n}$ matrices $B$ ${\ displaystyle B}$ :

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

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

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

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

For the block diagonal matrix

{\mbox{Pf}}{\begin{bmatrix}A_{1}&0\\0&A_{2}\end{bmatrix}}={\mbox{Pf}}(A_{1}){\mbox{Pf}}(A_{2}).

{\ displaystyle {\ mbox {Pf}} {\ begin {bmatrix} A_ {1} & 0 \\ 0 & A_ {2} \ end {bmatrix}} = {\ mbox {Pf}} (A_ {1}) {\ mbox {Pf}} (A_ {2}).}

For arbitrary $n\times n$ ${\ displaystyle n \ times n}$ matrices $M$ ${\ displaystyle M}$ :

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

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

History

The term “Pfaffian” was introduced by Cayley ^[1] and named after the German mathematician Johann Friedrich Pfaff .

Notes

↑ Earliest Known Uses of Some of the Words of Mathematics

Literature

Sluggish MN Pfaffians for enumeration problems // Summer School "Contemporary Mathematics". - 2004.
Sluggish MN Pfaffians or the art of placing signs ... // Mathematical Education . - 2005. - No. 9 .

[1] Earliest Known Uses of Some of the Words of Mathematics