Three nonnegativity questions for matrix powers

Exact statement

For a rational matrix $M$ , power nonnegativity asks whether $M^n\ge 0$ for some $n\ge 1$ ; eventual nonnegativity asks whether $M^n\ge 0$ for every sufficiently large $n$ ; everywhere nonnegativity asks whether this holds for every $n\ge 0$ . Inequalities are entrywise.

StatusPrecisely stated and used consistently

External reviewNo documented external or specialist review of this FPRD exposition is recorded.

Context

The three questions are mathematically different. In particular, allowing n=0 trivializes the existential question because M^0 is the identity.

Definitions

$\mathsf{PN}(M):\exists n\ge1\;M^n\ge0$ .
$\mathsf{ENN}(M):\exists N\;\forall n\ge N\;M^n\ge0$ .
$\mathsf{AN}(M):\forall n\ge0\;M^n\ge0$ .

Proof or evidence

The published matrix literature uses the universal-tail definition for eventual nonnegativity; the three displayed quantifier patterns are then formally distinct.

Three nonnegativity definitions →

Verification notes

The exponent-zero convention, entrywise order, and all three quantifier patterns were checked separately.

Limitations

This record fixes terminology; it is not an FPRD decision theorem.

Open work

Keep the exponent domain and quantifiers explicit in every later complexity or certificate statement.