Equivalence classes of inverse orthogonal and unit Hadamard matrices
Equivalence classes of inverse orthogonal and unit Hadamard matrices
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逆正交矩阵和单位 Hadamard 矩阵的等价类
DOI：
10.1017/s0004972700029506
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发表时间：
1991
影响因子：
0.7
通讯作者：
R. Craigen
中科院分区：
文献类型：
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作者：
R. Craigen
1867 年，Sylvester 考虑了 n × n 矩阵 (aij)，其具有非零复数值项，满足 (aij)(aij−1) = nI 他将这样的矩阵称为逆正交。如果逆正交矩阵的所有元素都在单位圆上，则它是单位哈达玛矩阵，并且具有通常意义上的正交性。任何两个逆正交（分别为单位哈达玛）矩阵是等价的，如果一个矩阵可以通过一系列涉及行和列的排列以及任何给定行或列中的所有条目乘以复数（分别为单位圆上的数字）的运算转换为另一个矩阵。他在没有证明的情况下指出，素数阶中恰好存在一个逆正交矩阵（因此也存在单位哈达玛矩阵）的等价类，并且一般来说，等价类的数量等于该阶的不同因式分解的数量。 1893 年，Hadamard 证明该断言在非素数阶单位 Hadamard 矩阵的情况下是错误的。我们为每个非素数阶和阶数 ≤ 3 给出正确的等价类数量，对于两种类型的矩阵，给出每个阶数 ≤ 4 的完整的、无冗余的类代表集。
In 1867, Sylvester considered n × n matrices, (aij), with nonzero complex-valued entries, which satisfy (aij)(aij−1) = nI Such a matrix he called inverse orthogonal. If an inverse orthogonal matrix has all entries on the unit circle, it is a unit Hadamard matrix, and we have orthogonality in the usual sense. Any two inverse orthogonal (respectively, unit Hadamard) matrices are equivalent if one can be transformed into the other by a series of operations involving permutation of the rows and columns and multiplication of all the entries in any given row or column by a complex number (respectively a number on the unit circle). He stated without proof that there is exactly one equivalence class of inverse orthogonal matrices (and hence also of unit Hadamard matrices) in prime orders and that in general the number of equivalence classes is equal to the number of distinct factorisations of the order. In 1893 Hadamard showed this assertion to be false in the case of unit Hadamard matrices of non-prime order. We give the correct number of equivalence classes for each non-prime order, and orders ≤ 3, giving a complete, irredundant set of class representatives in each order ≤ 4 for both types of matrices.