What is wrong with my argument using of Collatz–Wielandt formula?

Question

If $A$ is a positive square matrix, then the Collatz–Wielandt implies that

$\min_{=1,…,;_\neq 0}\frac{(Ay)_i}{y_i}≤≤\max_{=1,…,;_\neq 0}\frac{(Ay)_i}{y_i}$, Where $r$ is the largest eigenvalue of $A$.

By replacing $y=e_j$ in the previous expression wouldn't we obtain that $a_{jj}\leq r \leq a_{jj}$ for each $j\in\{1,\ldots,n\}$? This cannot be true for a matrix $A$ that has different arguments in the diagonal.

The previous inequality is from @Surb’s answer in here: Lower and upper bound for the largest eigenvalue.

Answer 1

No. In $e_j$, $j$ is not related to $i$. You would have $$ \min_i a_{ji} \leq r \leq \max_i a_{ji} $$ (... or possibly "$a_{ij}$" in both places, since I don't know whether you are using row first or column first indexing for matrices). You are pulling the minimum and maximum entries from the $j^\text{th}$ column.

Answer 2

Collatz-Wielandt theorm only states the lower bound. In fact, a counter example can be given by taking a matrix with large off diagonal entries.