Lower and upper bound for the largest eigenvalue We will call a matrix positive matrix if all elements in the matrix are positive, and we will denote the largest eigenvalue with $\lambda_{\max}$, what is exist because of the Perron–Frobenius theorem.
Theorem. Let $A$ be a positive square matrix. Then the minimal row sum is a lower bound and the maximal row sum is an upper bound of $\lambda_{\max}$.
My questions. 


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*Is there a name for this theorem and can anybody say books or papers what refer to it?

*How to prove it?
 A: Let us denote by $r$ the Perron-root of the positive matrix $A \in \mathbb{R}^{n \times n}$. Then by the Collatz-Wielandt formula we have:
$$\max_{x \in S}\min_{\substack{i=1, \ldots,n}} \frac{(Ax)_i}{x_i} = r = \min_{x \in S}\max_{\substack{i=1, \ldots,n}} \frac{(Ax)_i}{x_i}, $$
where $S := \{x \in \mathbb{R}^n\setminus\{0\}: x_i > 0, \forall i=1,\ldots,n\}$. Now it is clear that $A$ and $A^T$ have same eigenvalues, since $\det(M)=\det(M^T)$ and for every $\lambda \in \mathbb{R}$ we have $$\det(A-\lambda I)=\det((A-\lambda I)^T)= \det(A^T-\lambda I).$$
Furthermore $A$ strictly positive implies $A^T$ strictly positive, thus this formula also holds for $A^T$. It follows that we have
$$\max_{x \in S}\min_{\substack{i=1, \ldots,n}} \frac{(A^Tx)_i}{x_i} = r = \min_{x \in S}\max_{\substack{i=1, \ldots,n}} \frac{(A^Tx)_i}{x_i}.$$
This clearly implies that for every $y \in S$ we have
$$\min_{\substack{i=1, \ldots,n}} \frac{(A^Ty)_i}{y_i} \leq  r \leq \max_{\substack{i=1, \ldots,n}} \frac{(A^Ty)_i}{y_i}.$$
Choose $y = (1,1,\ldots,1)$ to get your bounds. Note also that using the same trick on $A$ directly you will get the same upper/lower bound but with the columns instead of the rows.
For reference, I recommend (in increasing order of technicality/generality):

*

*"Matrix analysis" by Horn and Johnson

*"Nonnegative Matrices in the Mathematical Sciences" by Bermann and Plemmons

*"Nonlinear Perron-Frobenius theory" by Nussbaum and Lemmens

There are even more general versions discussed in recent literature, e.g. in this paper
A: See propositions 2.3 and 2.4 of the book "Banach Lattices and Positive operators" of H.H. Schaefer.
