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CHARACTERISTIC POLYNOMIALS OF THE MATRICES WITH $(\,j,k)$-ENTRY $q^{\,j\pm k}+t$

Part of: Polynomials and matrices Basic linear algebra Sequences and sets Special matrices

Published online by Cambridge University Press:  03 June 2024

HAN WANG Open the ORCID record for HAN WANG [Opens in a new window]  and
ZHI-WEI SUN Open the ORCID record for ZHI-WEI SUN [Opens in a new window]
HAN WANG
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, PR China e-mail: hWang@smail.nju.edu.cn
ZHI-WEI SUN*
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, PR China
*
e-mail: zwsun@nju.edu.cn
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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 $\det [w_{\,j-k}+c\delta _{jk}]_{1\le \,j,k\le n}$.

Keywords

characteristic polynomialdeterminanteigenvalueLucas sequence

MSC classification

Primary: 11C20: Matrices, determinants
Secondary: 11B39: Fibonacci and Lucas numbers and polynomials and generalizations 15A15: Determinants, permanents, other special matrix functions 15A18: Eigenvalues, singular values, and eigenvectors 15B05: Toeplitz, Cauchy, and related matrices

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 111 , Issue 2 , April 2025 , pp. 228 - 237
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Supported by the National Natural Science Foundation of China (grant no. 12371004).

References

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Sun, Z.-W., ‘Sequences A355175 and A355326’, OEIS (On-Line Encyclopedia of Integer Sequences). Available at https://oeis.org.Google Scholar
Sun, Z.-W., Fibonacci Numbers and Hilbert’s Tenth Problem (Harbin Institute of Technology Press, Harbin, 2024) (in Chinese).Google Scholar