论文标题
在0--1矩阵上，其逆向具有同一模量的条目
On 0--1 matrices whose inverses have entries of the same modulus
论文作者
论文摘要
Barrett，Butler和Hall的猜想可能如下：如果$ n \ geq 3 $和$ a \ in \ {0,1 \}^{n \ times n} $（$ n \ times n $ 0- $ 0--1矩阵的家族）是一个非词的对称符号，则是以下两个陈述，然后是$ nike $ nike $ nikialtents $ nivalents n minivalents in n min n min。 $ n-2 $为零； （b）$ a^{ - 1} $是一个矩阵，其所有条目具有相同的模量，并且所有对角条目都相等。我们表明，如果$ a $没有零和非零的订单级别的$ n-4 $（如果$ n \ geq 5 $），则该猜想会保持。非对称矩阵$ a \ in \ {0,1 \}^{n \ times n} $的主要未成年人的奇偶校验，其订单$ n-2 $的主要未成年人都被探索了，尤其是探索了所有这些矩阵的决定因素。对于任意（不一定是对称的）非词组矩阵$ a \ in \ {0,1 \}^{n \ times n} $，使用$ n \ geq 3 $，我们确定$ a^{ - 1} $的必要条件，以使其成为一个矩阵的所有矩阵，其所有零件都具有相同的模量；此类条件的示例如下：$ a $的每个行和列都有偶数的非零条目； $ a^{ - 1} $的每个条目都是整数的倒数。 $ \ det（a）$偶数； $ a $的任何两个行之间的差异以及$ a $的任何两个列之间的差异都有偶数的非零条目。如果$ a $是对称的，则$ a $具有均匀数的非零对角条目；如果$ a $是对称的，而$ \ vec {a} _k $是$ a $的$ k $ th列，则$ a- \ vec {a} _k \ vec \ vec {a} _k^t $具有偶数的非零对角条目。
A conjecture of Barrett, Butler and Hall may be stated as follows: If $n \geq 3$ and $A \in \{0,1\}^{n \times n}$ (the family of $n \times n$ 0--1 matrices) is a nonsingular symmetric matrix, then the following two statements are equivalent: (a) All of the principal minors of $A$ of order $n-2$ are zero; and (b) $A^{-1}$ is a matrix all of whose entries have the same modulus and all of whose diagonal entries are equal. We show that this conjecture holds if $A$ does not have both a zero and a nonzero principal minor of order $n-4$ (if $n \geq 5$). The parity of the principal minors of nonsingular symmetric matrices $A \in \{0,1\}^{n \times n}$ whose principal minors of order $n-2$ are all zero is explored, establishing, in particular, that the determinants of such matrices are all even. For an arbitrary (not necessarily symmetric) nonsingular matrix $A \in \{0,1\}^{n \times n}$ with $n\geq 3$, we establish necessary conditions for $A^{-1}$ to be a matrix all of whose entries have the same modulus; examples of such conditions are the following: each row and column of $A$ has an even number of nonzero entries; each entry of $A^{-1}$ is the reciprocal of an even integer; $\det(A)$ is even; the difference between any two rows of $A$, as well as the difference between any two columns of $A$, has an even number of nonzero entries; if $A$ is symmetric, then $A$ has an even number of nonzero diagonal entries; if $A$ is symmetric and $\vec{a}_k$ is the $k$th column of $A$, then $A-\vec{a}_k\vec{a}_k^T$ has an even number of nonzero diagonal entries.
