论文标题
一类偏斜的对称toeplitz带矩阵的无效
Nullities for a class of skew-symmetric Toeplitz band matrices
论文作者
论文摘要
对于所有$ n> k \ ge 1 $，我们给出了$ n \ times n $ skew-skew-skew-skew-smemmetric toeplitz band matrix的nullity $ n（n，k）$的公式，其第一个$ k $ superdiagonals具有所有条目$ 1 $，其剩余的superdiagonals剩下的superdiagonals均为$ 0 $ 0。这是通过计算某些有向图中的循环数来完成的。作为一个应用程序，对于每个固定整数$ z \ ge 0 $和大型固定$ k $，我们给出了一个渐近公式，以满足$ n（n，k）= z $的$ n> k $的百分比。出于快速计算的目的，设计了一种算法，该算法快速计算$ n（n，k）$，即使对于极大的$ n $和$ k $的值。
For all $n > k \ge 1$, we give formulas for the nullity $N(n,k)$ of the $n \times n$ skew-symmetric Toeplitz band matrix whose first $k$ superdiagonals have all entries $1$ and whose remaining superdiagonals have all entries $0$. This is accomplished by counting the number of cycles in certain directed graphs. As an application, for each fixed integer $z\ge 0$ and large fixed $k$, we give an asymptotic formula for the percentage of $n > k$ satisfying $N(n,k)=z$. For the purpose of rapid computation, an algorithm is devised that quickly computes $N(n,k)$ even for extremely large values of $n$ and $k$.
