Theorem about positive matrices

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. If any element increases in the matrix then $\lambda_{\max}$ increases.

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?

Solution 1:

This is just a simple consequence of Perron-Frobenius theorem. We can prove something slightly more general:

Let $A,B$ be two nonnegative square matrices. If one of the following cases holds, then $\rho(A)<\rho(A+B)$:

• $A$ is irreducible (in particular, when $A>0$ entrywise) and $B$ is nonzero;
• $B$ is irreducible.

Let $(\rho(A),v)$ be a right Perron eigenpair of $A$ and $(\rho(A+B),w)$ be a left Perron eigenpair of the $A+B$. Then $$ \rho(A+B) w^Tv = w^T(A+B)v = \rho(A) w^Tv + w^TBv\tag{1} $$ and in turn $$ \left(\rho(A+B)-\rho(A)\right) w^Tv = w^TBv.\tag{2} $$ Now in case (a), both $v$ and $w$ positive (because $A$ and $A+B$ are irreducible). Hence $Bv$ is nonnegative but nonzero and $w^TBv,\ w^Tv$ are positive. So, $(2)$ gives $\rho(A+B)-\rho(A)>0$.

In case (b), $w>0$ because $A+B\ge0$ is irreducible. Hence $w^TB>0$ too because $B$ is irreducible. It follows that $w^TBv$ and $w^Tv$ are still positive (because $v$ is nonnegative but nonzero) and again, $(2)$ gives $\rho(A+B)-\rho(A)>0$.

Solution 2:

For (almost) all positive vectors $x$, the Power iterations do work, and we have $$\lambda_\max=\lim_{k\rightarrow\infty}\left(\|A^kx\|/\|x\|\right)^{1/k}$$ which shows that $\lambda_\max$ can not decrease as entries of $A$ increase. If $A'$ is obtained from $A$ by increasing every entry, we get $A'>(1+\varepsilon)A>A$, and your result follows.

Solution 3:

I recently came across the following (beautiful) result which is slightly improving the answer of @user1551. The following results come from the book Nonnegative Matrices in the Mathematical Sciences of Abraham Berman and Robert J. Plemmons:

Corollary 1.5 (p.27):
Let $A,B$ be nonnegative square matrices.
- If $0\leq A \leq B$, then $\rho(A)\leq \rho(B)$.
- If $0\leq A \leq B$, $A\neq B$ and $A+B$ is irreducible, then $\rho(A)<\rho(B)$.

Note 1: Here $A\leq B$ means $A_{i,j}\leq B_{i,j}$ for every $i,j$.

Note 2: If $0\leq A \leq B$, $A\neq B$ and $A+B$ is reducible, then both can occur, i.e. $\rho(A)=\rho(B)$ or $\rho(A)<\rho(B)$. Indeed, let $U,V,W\in\Bbb R^{2\times 2}$ with $$ U =\begin{pmatrix} 1& 0 \\ 0 & 1 \end{pmatrix}, \quad V =\begin{pmatrix} 1& 1 \\ 0 & 1 \end{pmatrix}, \quad W =\begin{pmatrix} 2 & 1 \\ 0 & 1 \end{pmatrix}.$$ Then $U\leq V \leq W$, $U\neq V$, $V\neq W$, $U+V$ is reducible, $V+W$ is reducible and $\rho(U)=\rho(V)<\rho(W)$.

Note 3: The following result shows that the answer of @user1551 is implied by Corollary 1.5.

Corollary 1.10 (p.28):
Let $A,B$ be nonnegative square matrices. If $A$ is irreducible, then $A+B$ is irreducible.

Finally, to see how this also improves the answer to your question, note that the matrix $V$ above is reducible as well as the matrix $$V'=\begin{pmatrix}0 & 0 \\ \epsilon & 0 \end{pmatrix}.$$ However, Corollary 1.5 implies that $\rho(V)<\rho(V+V')$ be cause $V+V'$ is strictly positive (and thus irreducible).

Solution 4:

Your property follows easily the Collatz-Wieland formula, which is proved on wikipedia. In fact, that wikipedia page proves everything from scratch and provides references. So it should contain everything you're asking for.

How to deduce your property from the Collatz-Wieland formula : let $A=(a_{ij})$ and $B=(b_{ij})$ be two positive matrices, such that $a_{ij} \leq b_{ij}$ for every $i,j$. Define the Collatz-Wieland functions

$$ f_A(x_1,x_2,\ldots,x_n)=\min_{1\leq i\leq n,x_i\neq 0}\frac{\sum_{j=1}^n a_{ij}x_j}{x_i}, \ f_B(x_1,x_2,\ldots,x_n)=\min_{1\leq i\leq n,x_i\neq 0}\frac{\sum_{j=1}^n b_{ij}x_j}{x_i} $$

We deduce $f_A(x)\leq f_B(x)$ for every $x$, so the Perron root of $A$ is smaller than the Perron root of $B$, by the Collatz-Wieland formula.