Suppose $\mu_1 \leqslant \mu_2 \leqslant \cdots \leqslant \mu_n$ be the eigenvalues of the matrix $A$ and $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$ be the eigenvalues of the matrix $A -\mathbb{E}A$. Here $\mathbb{E}A$ has $(i,j)^{th}$ entry as expectation of the $(i,j)^{th}$ entry of $A$. Need to show that the interlacing property that $\lambda_1 \leqslant \mu_1 \leqslant \lambda_2 \leqslant \mu_2 \leqslant \cdots \leqslant \lambda_n \leqslant \mu_n$.
I was going through a paper. There the authors conclude the interlacing property by the following equation. $$\left(\sum_{i=1}^{n}\frac{x_i^2}{\mu_i-z}\right)^{-1} = f + \left(\sum_{i=1}^{n}\frac{y_i^2}{\lambda_i-z}\right)^{-1}$$ where $x_i,y_i$ are real numbers and $z$ is the variable. They say, that putting $z=\lambda_i$ for $i=1,2, \cdots, n$ gives the conclusion.
For $n=1$, it is easy to conclude. For $n=2$ also gives a simpler equation but I can not conclude the interlacing property. It is more hard to understand for $n \geqslant 3$.
Any further help is much appreciated.
Thanks in advance.