论文标题
与Pascal的三角形和斐波那契序列连接的三角形真实对称矩阵
Tridiagonal real symmetric matrices with a connection to Pascal's triangle and the Fibonacci sequence
论文作者
论文摘要
我们探索某个家庭$ \ {a_n \} _ {n = 1}^{\ infty} $ $ n \ times n $ tridiagonal真实对称矩阵。在得出该家族特征多项式的三届复发关系之后，我们找到了封闭形式的解决方案。这些特征多项式的系数证明涉及帕斯卡尔三角形的对角线，以诱人的预测方式。最后，我们探讨了家庭各个成员的特征值之间的关系。更具体地说，我们在$ \ texttt {spec}（a_m）$中包含在$ \ texttt {spec}（a_n）$中的$ \ texttt {spec}（a_m）$时，对$ m，n \ in \ mathbb {n} $的值给出了足够的条件。我们以许多开放的问题结束了论文，其中之一将我们的特征多项式与斐波那契序列相结合，涉及椭圆的有趣方式。
We explore a certain family $\{A_n\}_{n=1}^{\infty}$ of $n \times n$ tridiagonal real symmetric matrices. After deriving a three-term recurrence relation for the characteristic polynomials of this family, we find a closed form solution. The coefficients of these characteristic polynomials turn out to involve the diagonal entries of Pascal's triangle in a tantalizingly predictive manner. Lastly, we explore a relation between the eigenvalues of various members of the family. More specifically, we give a sufficient condition on the values $m,n \in \mathbb{N}$ for when $\texttt{spec}(A_m)$ is contained in $\texttt{spec}(A_n)$. We end the paper with a number of open questions, one of which intertwines our characteristic polynomials with the Fibonacci sequence in an intriguing manner involving ellipses.
