$\forall \lambda \in \mathcal{F}: A x=\lambda x: \lambda \in \mathbb{R}$ - Correct notation?

Question

I am new to formal mathematics, and thus not very experienced with correct notation. Consider $A\in End(V)$ where $V$ is a vector space over $\mathcal{F}$. I want to write $\sigma(A)\subseteq \mathbb{R}$ more explicitly, also using symbols. Would this be correct: $$ \forall \lambda \in \mathcal{F}: A x=\lambda x: \lambda \in \mathbb{R} $$ In words, I want to say that all eigenvalues of $A$ are real.

Answer 1

I would approach this issue by starting with words and working towards symbols. Also, I would work with one portion at a time, starting by carefully expressing the spectrum of $A$ using set builder notation:

• The spectrum of $A$ is set of all elements of $\mathcal F$ that are eigenvalues of $A$.
• $\sigma(A)= \{\lambda \in \mathcal F : \lambda$ is an eigenvalue of $A \}$.
• $\sigma(A) = \{\lambda \in \mathcal F : \text{there exists $x \in V - \{0\}$ such that $Ax = \lambda x$} \}$
• $\sigma(A) = \{\lambda \in \mathcal F : \exists x \in V - \{0\} \, \text{such that} \, Ax = \lambda x \}$

One could go further and replace the words "such that" with some symbols, if one truly desired.

Now to express that $\sigma(A) \subset \mathbb R$ you can just substitute

$\{\lambda \in \mathcal F : \exists x \in V - \{0\} \, \text{such that} \, Ax = \lambda x \} \subset \mathbb R$

But, I have two followup thoughts:

1. The elements a general field $\mathcal F$ need not be real numbers, so this well expressed statement might be completely false. Of course, in some cases it becomes an interesting statement, particularly when $\mathcal F$ is the field of complex numbers.
2. I realize that you are more-or-less just trying to understand how to write good notation, but keep in mind that there are good and bad reasons to express things in notation. If I had to write this statement in a paper for purposes of communicating mathematics to a reader who is a human being, I would write "the eigenvalues of $A$ are real numbers".