Effects of increasing a matrix's values on the eigenvalue decomposition Let $A \in \mathbb{R}^{n \times n}$ be a real, symmetric positive semi-definite matrix, and let $A = UVU^T$ be $A$'s eigendecomposition.
Suppose that the matrix $A'$ was obtained by $A$ by making some values of $A$ larger, in a way that $A'$ is still symmetric and positive semi-definite matrix.
Can we say something about $A'$ eigendecomposition in terms of $A$'s eigendecomposition? In particular, does $A'$ eigenvalues necessarily larger?   

A very simple example for some n x n matrix  
consider  
$B(\tau) = \tau \cdot2\mathbf{11}^T + (1-\tau) \cdot I$
for $\tau \in [0,1]$
at
$B(0)=I $
$B(1)=\mathbf{11}^T $ 
the dominant eigenvalue (perron root) $\lambda_1$  icreases with $\tau$ and all other eigenvalues decrease as $\tau$ grows.  Yet all matrix components increase as $\tau$ increases.
