Interpretation of $\mathbb{R}^n/\mathbb{R}\mathrm{1}$

Question

Not exactly sure how to read $\mathbb{R}^n/\mathbb{R}\mathrm{1}$ where $\mathrm{1}=(1,1,...,1)$ is the unit vector. Here's the context:

We next determine the eigenspace of the matrix $\textit{A}$, which is the set $$Eig(A) = \{x\in\mathbb{R}^n:A \odot x = \lambda (A) \odot x \}$$ Clearly, Eig(A) is closed under tropical scalar multiplication, that is, if $x\in$Eig($A$) and $c\in \mathbb{R}$ then $c\odot x$ is also in Eig($A$). We can thus identify Eig($A$) with its image in $\mathbb{R}^n/\mathbb{R}\mathrm{1}\simeq\mathbb{R}^{n-1}$

Do I read this set as being "$\mathbb{R}^n$ modulo some real-valued multiples of the unit vector"? What references are available for me to refer to in order to get some clarity on how such a set would be isomorphic to $\mathbb{R}^{n-1}$?

EDIT:

A $\textit{tropical algebra}$ can be defined as a semi-field over the real numbers with the following operations: $$ x\oplus y:= \max\{x,y\}$$ $$ x\odot y := x+y$$

Matrix multiplication by other matrices and by scalars follow from these definitions as one would expect.

Answer 1

$\mathbb R^n / \mathbb R 1$ is a quotient vector space. This is a standard construction in linear algebra. You can look at it as the set of equivalence classes of elements of $\mathbb R^n$ where $a \sim b$ if $a = b + k1$ for some $k \in \mathbb R$. This has a natural vector space structure, but that is not really the structure in use here. This passage looks like it's defining tropical eigenvalues; tropical scalar multiplication is the same as regular addition. That is, $c \odot x = c1 + x$. This means that each tropical eigenvector $x$ of $A$ has an 'eigenspace' $\{c \odot x : c \in \mathbb R\}$, which is precisely the equivalence class associated with $x$ in $\mathbb R^n / \mathbb R 1$. Technically speaking, what the passage says is not exactly true: $\text{Eig}(A)$ is not identified with its image in $\mathbb R^n/\mathbb R 1$, but the quotient of $\text{Eig}(A)$ by tropical scalar multiplication is.

I will add that it is a bit strange that you seem to be studying tropical linear algebra without an understanding of quotient spaces. Perhaps you would be best served by learning linear algebra on the level of the book by Axler before moving on to more exotic structures.

Answer 2

$\mathbf{R}^n/\mathbf{R}\pmb 1$ is a tropical analogue of projective space. I.e. it is the set of vectors $(a_1,\dots,a_n) \in \mathbf{R}^n$ module the relation

$$ (a_1,\dots,a_n) \sim (\lambda \odot a_1,\dots,\lambda \odot a_n)\quad, \forall \lambda \in \mathbf R. $$

In classical notation this is just $$ v \sim v + \lambda \pmb 1$$

which is exactly the vector space quotient of $\mathbf{R}^n$ by $\mathbf{R}\pmb 1 = \operatorname{span}\{\pmb 1\}$.