How to construct a upper triangular matrix that satisfy the quadratic polynomial equation with different roots

TL;DR

This paper characterizes all upper triangular matrices over a ring that satisfy a specific quadratic polynomial with distinct roots, providing explicit descriptions and counting solutions in finite rings.

Contribution

It offers a complete description of matrices satisfying quadratic equations with distinct roots over rings, including counting solutions in finite rings.

Findings

Explicit characterization of matrices satisfying quadratic equations over rings.

Counting solutions in finite rings for the matrix quadratic equation.

Extension to infinite upper triangular matrices.

Abstract

Let $R$ be an associative ring with identity $1$. We describe all matrices in $T_n(R)$ the ring of $n\times n$ upper triangular matrices over $R$ ($n\in \mathbb{N}$), and $T_{\infty}(R)$ the ring of infinite upper triangular matrices over $R$, satisfying the quadratic polynomial equation $x^2-rx+s=0$. For such propose we assume that the above polynomial have two different roots in $R$. Moreover, in the case that $R$ in finite, we compute the number of all matrices to solves the matrix equation $A^2-rA+sI=0,$ where $I$ is the identity matrix.