<s>[INST] Matrix rescaling increases lowest eigenvalue? Consider the set $\mathbf{N}:=\left\{1,2,....,N \right\}$ and let $$\mathbf M:=\left\{ M_i; M_i \subset \mathbf N \text{ such that } \left\lvert M_i \right\rvert=2 \text{ or }\left\lvert M_i \right\rvert=1 \right\}$$
be the set of all subsets of $\mathbf{N}$ that are of cardinality $1$ or $2.$
The cardinality of the set $\mathbf M$ itself is $\binom{n}{1}+\binom{n}{2}=:K$
We can then study for $y \in (0,1)$ the $K \times K$ matrix
$$A_N = \left(  \frac{\left\lvert M_i \cap M_j \right\rvert}{\left\lvert M_i \right\rvert\left\lvert M_j \right\rvert}y^{-\left\lvert M_i \cap M_j \right\rvert} \right)_{i,j}$$
and 
$$B_N = \left(  \left\lvert M_i \cap M_j \right\rvert y^{-\left\lvert M_i \cap M_j \right\rvert} \right)_{i,j}.$$
Question
I conjecture that $\lambda_{\text{min}}(A_N)\le \lambda_{\text{min}}(B_N)$ for any $N$ and would like to know if one can actually show this?
As a first step, I would like to know if one can show that $$\lambda_{\text{min}}(A_N)\le C\lambda_{\text{min}}(B_N)$$ for some $C$ independent of $N$?
In fact, I am not claiming that $A_N \le B_N$ in the sense of matrices. But it seems as if the eigenvalues of $B_N$ are shifted up when compared with $A_N.$
Numerical evidence:
For $N=2$ we can explicitly write down the matrices 
$$A_2 =\left(
\begin{array}{ccc}
 \frac{1}{y} & 0 & \frac{1}{2 y} \\
 0 & \frac{1}{y} & \frac{1}{2 y} \\
 \frac{1}{2 y} & \frac{1}{2 y} & \frac{1}{2 y^2} \\
\end{array}
\right) \text{ and }B_2 = \left(
\begin{array}{ccc}
 \frac{1}{y} & 0 & \frac{1}{y} \\
 0 & \frac{1}{y} & \frac{1}{y} \\
 \frac{1}{y} & \frac{1}{y} & \frac{2}{y^2} \\
\end{array}
\right)$$
We obtain for the lowest eigenvalue of $A_2$ (orange) and $B_2$(blue) as a function of $y$

For $N=3$ we get qualitatively the same picture, i.e. the lowest eigenvalue of $A_3$ remains below the lowest one of $B_3$:
In this case: 
$$A_3=\left(
\begin{array}{cccccc}
 \frac{1}{y} & 0 & 0 & \frac{1}{2 y} & 0 & \frac{1}{2 y} \\
 0 & \frac{1}{y} & 0 & \frac{1}{2 y} & \frac{1}{2 y} & 0 \\
 0 & 0 & \frac{1}{y} & 0 & \frac{1}{2 y} & \frac{1}{2 y} \\
 \frac{1}{2 y} & \frac{1}{2 y} & 0 & \frac{1}{2 y^2} & \frac{1}{4 y} & \frac{1}{4 y} \\
 0 & \frac{1}{2 y} & \frac{1}{2 y} & \frac{1}{4 y} & \frac{1}{2 y^2} & \frac{1}{4 y} \\
 \frac{1}{2 y} & 0 & \frac{1}{2 y} & \frac{1}{4 y} & \frac{1}{4 y} & \frac{1}{2 y^2} \\
\end{array}
\right)\text{ and } B_3=\left(
\begin{array}{cccccc}
 \frac{1}{y} & 0 & 0 & \frac{1}{y} & 0 & \frac{1}{y} \\
 0 & \frac{1}{y} & 0 & \frac{1}{y} & \frac{1}{y} & 0 \\
 0 & 0 & \frac{1}{y} & 0 & \frac{1}{y} & \frac{1}{y} \\
 \frac{1}{y} & \frac{1}{y} & 0 & \frac{2}{y^2} & \frac{1}{y} & \frac{1}{y} \\
 0 & \frac{1}{y} & \frac{1}{y} & \frac{1}{y} & \frac{2}{y^2} & \frac{1}{y} \\
 \frac{1}{y} & 0 & \frac{1}{y} & \frac{1}{y} & \frac{1}{y} & \frac{2}{y^2} \\
\end{array}
\right)$$

 [/INST] 
Claim. $\lambda_\min(A_N) \le 4\lambda_\min(B_N)$.

Proof.
Let $C_N:=\bigl[\tfrac{1}{|M_i||M_j|}\bigr]$. Then, $B_N = A_N \circ C_N$, where $\circ$ denotes the Hadamard product. Observe that by construction both $A_N$ and $C_N$ are positive semidefinite, so $B_N$ is also psd. Let's drop the subscript $N$ for brevity. Define $c = \text{diag}(C)$ sorted in decreasing order, so in particular, $c_\min = \min_{1\le i \le N} 1/|M_i|^2=1/4$.
Now, from Theorem 3(ii) of Bapat and Sunder, it follows that:
\begin{equation*}
\lambda_\min(B)=\lambda_\min(A \circ C) \ge \lambda_\min(A)c_\min = \lambda_\min(A_N)/4.
\end{equation*}

Note: The result of Bapat and Sunder is more general. For psd matrices $A$ and $C$ it states that
\begin{equation*}
\prod_{j=k}^n \lambda_j(A\circ C) \ge \prod_{j=k}^n\lambda_j(A)c_j,
\end{equation*}
where $1\le k \le n$, and $\lambda_1(\cdot)\ge \lambda_2(\cdot) \ge \cdots \ge \lambda_n(\cdot)$, while $c$ is as above.
</s>