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CHARACTERISTIC POLYNOMIALS OF THE MATRICES WITH
$(\,j,k)$-ENTRY
$q^{\,j\pm k}+t$
Published online by Cambridge University Press: 03 June 2024
Abstract
We determine the characteristic polynomials of the matrices $[q^{\,j-k}+t]_{1\le \,j,k\le n}$ and
$[q^{\,j+k}+t]_{1\le \,j,k\le n}$ for any complex number
$q\not =0,1$. As an application, for complex numbers
$a,b,c$ with
$b\not =0$ and
$a^2\not =4b$, and the sequence
$(w_m)_{m\in \mathbb Z}$ with
$w_{m+1}=aw_m-bw_{m-1}$ for all
$m\in \mathbb Z$, we determine the exact value of
$\det [w_{\,j-k}+c\delta _{jk}]_{1\le \,j,k\le n}$.
MSC classification
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
Supported by the National Natural Science Foundation of China (grant no. 12371004).