Tsukuba Journal of Mathematics

On some matrix diophantine equations

Aleksander Grytczuk and Izabela Kurzydlo

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Abstract

Let $A \in M_n(\mathbf{C})$, $n \ge 2$ be the matrix which has at least one real eigenvalue $\alpha \in (0, 1)$. If the matrix equation \begin{equation} A^x + A^y + A^z = A^w \tag{1} \end{equation} is satisfied in positive integers $x$, $y$, $z$, $w$, then $\max \{x-w, y-w, z-w\} \ge 1$. If suppose that the matrix $A$ has at least one real eigenvalue $\alpha > \sqrt{2}$ and the equation (1) is satisfied in positive integers $x$, $y$, $z$ and $w$, then $\max \{x-w, y-w, z-w\} = -1$. Moveover, we investigate the solvability of the matrix equations (1) and \begin{equation} A^x + A^y = A^z \tag{2} \end{equation} for the non-negative real $n \times n$ matrices, where $|\det A| > 1$, in positive integers $x$, $y$, $z$, $w$ for (1) and $x$, $y$, $z$ for (2). Using the wellknown theorem of Perron-Frobenius we obtain some informations concerning solvability these equations.

Article information

Source
Tsukuba J. Math., Volume 33, Number 2 (2009), 299-304.

Dates
First available in Project Euclid: 26 February 2010

Permanent link to this document
https://projecteuclid.org/euclid.tkbjm/1267209422

Digital Object Identifier
doi:10.21099/tkbjm/1267209422

Mathematical Reviews number (MathSciNet)
MR2605857

Zentralblatt MATH identifier
1242.15012

Subjects
Primary: 15A24: Matrix equations and identities 15A42: Inequalities involving eigenvalues and eigenvectors

Keywords
The matrix equations Schur's Lemma Fermat's type equation on matrices

Citation

Grytczuk, Aleksander; Kurzydlo, Izabela. On some matrix diophantine equations. Tsukuba J. Math. 33 (2009), no. 2, 299--304. doi:10.21099/tkbjm/1267209422. https://projecteuclid.org/euclid.tkbjm/1267209422


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