What is the pfaffian of $n\times n$ antisymmetric matrix whose all upper diagonal entries are 1? Is there any method to compute?
2026-02-22 19:55:17.1771790117
Pfaffian of the antisymmetric matrix whose all upper diagonal entries are 1
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I suppose you mean that the strictly upper triangular part of the skew-symmetric matrix $A$ contains only ones. If we define a matrix $P$ as follows, then $P^TAP$ has a very nice form: $$ P=\pmatrix{1&-1\\ &\ddots&\ddots\\ &&\ddots&-1\\ &&&1}, \quad P^TAP=\pmatrix{0&1\\ -1&\ddots&\ddots\\ &\ddots&\ddots&1\\ &&-1&0}. $$ You may now obtain the value of $\operatorname{pf}(A)$ easily by using some well-known properties of Pfaffian (see Wikipedia for some such properties).