两道特殊的三对角行列式
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*} 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. [i=s] 本帖最后由 青青子衿 于 2019-4-24 14:30 编辑 [/i]
[quote]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. ...
[size=2][color=#999999]orzweb111 发表于 2019-4-24 00:39[/color] [url=http://kuing.orzweb.net/redirect.php?goto=findpost&pid=30993&ptid=6043][img]http://kuing.orzweb.net/images/common/back.gif[/img][/url][/size][/quote]
谢谢！找到一篇相关文章了。
[i]Another look at a matrix of Mark Kac[/i]
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
\(\,\left|\boldsymbol{A}_n-\lambda\boldsymbol{E}\right|=D_{a,n}({\color{red}-}\lambda)\,\)
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