In the course of a big perturbation calculation I was doing, a particular matrix ring kept popping up, whose nice properties were essential to being able to complete the calculation. As such, I've become curious what this group is, and what else is known about it.
The ring consists of all matrices of the form $$a I + b N$$ where $a,b\in \mathbb{R}$, $a\neq 0$, $I$ is the $n\times n$ identity matrix, and $N$ is the $n\times n$ matrix of all ones. A few basic facts I've worked out about such matrices:
- $(a I + b N)(c I +d N)= acI + (bc+ad+bd n)N$, which is true because $N^2 = nN$.
- $\det(a I +b N)=a^{n-1}(a+bn)$. This fact can then easily be used to compute the eigenvalues of $aI+bN$.
- $(aI+bN)^{-1}=\frac{1}{nab+a^2}\left( (a+nb)I-bN \right )$ so long as $a \neq -nb$.
- The ring is commutative.
Given that these matrices are so computationally tractable, I feel like they must have been studied, and must have a name. Does anybody know what this ring is? Thank you in advance.