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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.

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

  2. How to prove it?

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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:

$$\max_{x \in S}\min_{\substack{i=1, \ldots,n\\ x_i \neq 0}} \frac{(Ax)_i}{x_i} = r = \min_{x \in S}\max_{\substack{i=1, \ldots,n\\ x_i \neq 0}} \frac{(Ax)_i}{x_i}, $$ where $S:= \{x \in \mathbb{R}^n\setminus\{0\}: x_i \geq 0, \forall i=1,\ldots,n\}$. See here page 667/668 for a reference. 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 $$\max_{x \in S}\min_{\substack{i=1, \ldots,n\\ x_i \neq 0}} \frac{(A^Tx)_i}{x_i} = r = \min_{x \in S}\max_{\substack{i=1, \ldots,n\\ x_i \neq 0}} \frac{(A^Tx)_i}{x_i}.$$ This clearly implies that for every $y \in S$ we have $$\min_{\substack{i=1, \ldots,n\\ y_i \neq 0}} \frac{(A^Ty)_i}{y_i} \leq r \leq \max_{\substack{i=1, \ldots,n\\ y_i \neq 0}} \frac{(A^Ty)_i}{y_i}.$$ 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.

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See propositions 2.3 and 2.4 of the book "Banach Lattices and Positive operators" of H.H. Schaefer.

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  • $\begingroup$ Can you maybe offer a good ebook version of it, or send me an image about the pages what you talking about? It could really help me. $\endgroup$ – user153012 Aug 26 '14 at 21:05
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    $\begingroup$ Or just write it here. $\endgroup$ – user153012 Aug 26 '14 at 21:21
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    $\begingroup$ See this link. books.google.com.br/…. I could not find a better look inside this book. I forgot to say that the page is 7. The number $r(A)$ is the spectral radius of $A$, which is the biggest positive eigenvalue for this type of matrix. $\endgroup$ – Daniel Aug 27 '14 at 1:25

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