Let $P \in \rm{GL}\_N(\mathbb{C})$. Call the columns of $P$ $|R\_i\rangle$ and the rows of $P^{-1}$ $\langle L\_i|$, so that
$$
\langle L\_i | R\_j \rangle = \delta\_{i,j}.
$$ 
And define the matrix $G$ by 
$$
G\_{i,j} = \langle R\_i | R\_j \rangle \langle L\_j | L\_i \rangle.
$$
One can prove that $G$ is hermitian and positive-definite, and also that for any $i$,
$$
\sum\_{j=1}^N G\_{i,j} = 1,
$$
which implies that $1 \in \rm{Spec}(G)$. I would like to know whether $1$ is always the **smallest** eigenvalue. It seems to be the case but I have no proof of this fact.



Eigenvalues of a generalized Gram matrix
If I understand correctly, in conventional mathematical notation this would be the Hadamard product $X \circ X^{-T}$ (I think it's a pure transpose and not a conjugate transpose, but please double-check this) with the positive definite matrix $X = P^\*P$. So this assertion is a consequence of Theorem 5.4.3 in Horn and Johnson's *Topics in Matrix Analysis*:



> 
> Let $A \in M\_n$ be positive definite. Then
> 
> 
> (a) $\lambda\_{\min}(A \circ A^{-1}) \geq 1$
> 
> 
> (b) $\lambda\_{\min}(A \circ A^{-T}) = 1$
> 
> 
> 


