Lower and upper bound for the largest eigenvalue
Mia Lopez 
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.
Is there a name for this theorem and can anybody say books or papers what refer to it?
How to prove it?
2 Answers
$\begingroup$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
$\endgroup$ 6 $\begingroup$See propositions 2.3 and 2.4 of the book "Banach Lattices and Positive operators" of H.H. Schaefer.
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