Let be {\ displaystyle \ Pi} denotes the set of all partitions of the set {\ displaystyle \ {1,2, \ dots, 2n \}} unordered pairs (total exists {\ displaystyle (2n-1) !!} such partitions). Breaking up {\ displaystyle \ alpha \ in \ Pi} can be recorded
- {\ displaystyle \ alpha = \ {(i_ {1}, j_ {1}), (i_ {2}, j_ {2}), \ cdots, (i_ {n}, j_ {n}) \}}
Where {\ displaystyle i_ {k} <j_ {k}} and {\ displaystyle i_ {1} <i_ {2} <\ cdots <i_ {n}} . Let be
- {\ 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 {\ displaystyle {\ mbox {sgn}} (\ alpha)} - permutation sign {\ displaystyle \ pi} . It is easy to see that {\ displaystyle {\ mbox {sgn}} (\ alpha)} independent of choice {\ displaystyle \ pi} .
Let be {\ displaystyle A = \ {a_ {ij} \}} denotes {\ displaystyle 2n \ times 2n} skew-symmetric matrix. For splitting {\ displaystyle \ alpha} define
- {\ 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
- {\ displaystyle \ operatorname {Pf} (A) = \ sum _ {\ alpha \ in \ Pi} A _ {\ alpha}.}
Pfaffian skew-symmetric matrix size {\ displaystyle n \ times n} for odd n is equal to zero by definition.
Recursive Definition
Pfaffian size matrix {\ displaystyle 0 \ times 0} set equal to 1; Pfaffian of a skew-symmetric matrix A of size {\ displaystyle 2n \ times 2n} at {\ displaystyle n> 0} can be defined recursively as follows:
- {\ 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 {\ displaystyle i} can be chosen arbitrarily {\ displaystyle \ theta (ij)} - Heaviside function , {\ displaystyle A _ {{\ hat {\ imath}} {\ hat {\ jmath}}}} denotes a matrix A without the ith and jth columns and rows.
Alternative Definition
For {\ displaystyle 2n \ times 2n} skew-symmetric matrix {\ displaystyle A = \ {a_ {ij} \}} consider the bivector :
- {\ displaystyle \ omega = \ sum _ {i <j} a_ {ij} \; e_ {i} \ wedge e_ {j}.}
Where {\ displaystyle \ {e_ {1}, e_ {2}, \ dots, e_ {2n} \}} there is a standard basis in {\ displaystyle \ mathbb {R} ^ {2n}} . Then Pfaffian is defined by the following equation:
- {\ displaystyle {\ frac {1} {n!}} \ omega ^ {\ wedge n} = {\ mbox {Pf}} (A) \; e_ {1} \ wedge e_ {2} \ wedge \ dots \ wedge e_ {2n},}
Where {\ displaystyle \ omega ^ {\ wedge n}} denotes the outer product of n copies {\ displaystyle \ omega} .
For {\ displaystyle 2n \ times 2n} skew-symmetric matrix {\ displaystyle A} and for arbitrary {\ displaystyle 2n \ times 2n} matrices {\ 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 the block diagonal matrix
- {\ displaystyle {\ mbox {Pf}} {\ begin {bmatrix} A_ {1} & 0 \\ 0 & A_ {2} \ end {bmatrix}} = {\ mbox {Pf}} (A_ {1}) {\ mbox {Pf}} (A_ {2}).}
- For arbitrary {\ displaystyle n \ times n} matrices {\ displaystyle M} :
- {\ displaystyle {\ mbox {Pf}} {\ begin {bmatrix} 0 & M \\ - M ^ {T} & 0 \ end {bmatrix}} = (- 1) ^ {n (n-1) / 2} \ det M.}
The term “Pfaffian” was introduced by Cayley [1] and named after the German mathematician Johann Friedrich Pfaff .