About non-negative matrix

Question

If $W$ is diagonal matrix with each entry $W_{i,i}>0$, $K$ is a symmetric and positive semi-definite matrix and $K_{i,j}>0$ (actually $K$ is a kernel matrix and calculated from a RBF kernel function), each element of vector $x$ is larger than 0, can we say that $(W+K^{-1})^{-1}x$ is a vector with each entry larger than 0?

Answer 1

You are essentially asking whether $(W+K^{-1})^{-1}$ is a (entrywise) positive matrix. The answer is negative. Counterexample: suppose $W=10\,I_3$, $$ X=(x_1,x_2,x_3)=\begin{pmatrix} 0.9883 &0.5155 &0.2259\\ 0.5828 &0.3340 &0.5798\\ 0.4235 &0.4329 &0.7604 \end{pmatrix} $$ and $K$ is the RBF kernel matrix such that $K_{ij}=\exp(-\|x_i-x_j\|_2^2)$. Then $$ (W+K^{-1})^{-1}=\begin{pmatrix} 0.0829 &0.0124 &-0.0010\\ 0.0124 &0.0728 &0.0136\\ -0.0010 &0.0136 &0.0818 \end{pmatrix}. $$ Therefore, when $x=(1,\varepsilon,\varepsilon)^T$ for sufficiently small $\varepsilon>0$, the third entry of $(W+K^{-1})^{-1}x$ is negative.