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In mathematics, the determinant of a skew-symmetric matrix can always be written as the square of a polynomial in the matrix entries. This polynomial is called the Pfaffian of the matrix. The term ''Pfaffian'' was introduced by who named them after Johann Friedrich Pfaff. The Pfaffian is nonvanishing only for 2''n'' × 2''n'' skew-symmetric matrices, in which case it is a polynomial of degree ''n''. Explicitly, for a skew-symmetric matrix A, : which was first proved by Thomas Muir in 1882 . The fact that the determinant of any skew symmetric matrix is the square of a polynomial can be shown by writing the matrix as a block matrix, then using induction and examining the Schur complement, which is skew symmetric as well. 〔Ledermann, W. "A note on skew-symmetric determinants"〕 ==Examples== : : (3 is odd, so Pfaffian of B is 0) : The Pfaffian of a 2''n'' × 2''n'' skew-symmetric tridiagonal matrix is given as : (Note that any skew-symmetric matrix can be reduced to this form with all equal to zero, see Spectral theory of a skew-symmetric matrix) 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「pfaffian」の詳細全文を読む スポンサード リンク
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