两道特殊的三对角行列式

青青子衿

三对角行列式为\(D_{a,n}(x)\)
\[D_{a,n}(x)=\left|\begin{array}{ccccccc}
x&1&&&&&\\
n-1&x&2&&&&\\
&n-2&x& \ddots &&&\\
&& \ddots & \ddots & \ddots && \\
&&& \ddots &x&n-2&\\
&&&&2&x&n-1\\
&&&&&1&x
\end{array}\right|\]
三对角行列式为\(D_{b,n}(x)\)
\[D_{b,n}(x)=\left|\begin{array}{ccccccc}
x&1&&&&&\\
-(n-1)&x&2&&&&\\
&-(n-2)&x& \ddots &&&\\
&& \ddots & \ddots & \ddots && \\
&&& \ddots &x&n-2&\\
&&&&-2&x&n-1\\
&&&&&-1&x
\end{array}\right|\]
\begin{gather*}
&D_{a,n}(x)&=&\begin{cases}
(x^2-1^2)\cdots\left(x^2-(n-3)^2\right)\left(x^2-(n-1)^2\right)&n= 0\pmod2\\
x(x^2-2^2)\cdots\left(x^2-(n-3)^2\right)\left(x^2-(n-1)^2\right)& n= 1\pmod2 \end{cases}\\
\,\\
&D_{b,n}(x)&=&\begin{cases}
(x^2+1^2)\cdots\left(x^2+(n-3)^2\right)\left(x^2+(n-1)^2\right)&n= 0\pmod2\\
x(x^2+2^2)\cdots\left(x^2+(n-3)^2\right)\left(x^2+(n-1)^2\right)& n= 1\pmod2 \end{cases}\\
\end{gather*}

orzweb111

The entry $x$ in the diagonal doesn't play much role, so one usually igores it.  $D_{a,n}(0)$ is usually known as a Matrix of Mark Kac in the literature.

青青子衿

The entry $x$ in the diagonal doesn't play much role, so one usually igores it.  $D_{a,n}(0)$ is usually known as a Matrix of Mark Kac in the literature. ...
orzweb111 发表于 2019-4-24 00:39

谢谢！找到一篇相关文章了。
Another look at a matrix of Mark Kac
Olga Taussky, JohnTodd
Linear Algebra and its Applications
Volume 150, May 1991, Pages 341-360

实际上，可以将\(\,D_{a,n}({\color{red}-}\lambda)\,\)看作是Mark Kac矩阵\(\,\boldsymbol{A}_{n}\,\)的特征多项式
\(\,\left|\boldsymbol{A}_n-\lambda\boldsymbol{E}\right|=D_{a,n}({\color{red}-}\lambda)\,\)

hbghlyj

该帖子已被页面一个特殊的三对角形行列式求解引用