Recently i'm reading a paper,there is a inequality that confuse me. L is a symmetric,irreducible and semi-positive definite matrix with eigenvalues of $0=\lambda\_{1}(L)<\lambda\_{2}(L)\leq...\leq\lambda\_{m}(L)$(counting the multiplicities),P is a positive definite matrix with maximum eigenvalues $\lambda\_{m}(P)$,both of them are square matrix,and it satisfied


$$x^{T}(L^{2}\otimes P^{2})x\leq \lambda\_{m}(L)\lambda\_{m}(P)x^{T}(L\otimes P)x$$


I don't know how to prove this inequality,I have searched many references ,but hardly find any result.Maybe there needs other properties,welcome to point out!



an inequality about kronecker product with eigenvalues question
We use $L^t \otimes P^t = (L\otimes P)^t$ twice below. The proof follows by observing that
\begin{eqnarray\*}
\sup\_{x\neq 0}\frac{x^T(L\otimes P)^2 x}{x^T(L\otimes P)x} = \sup\_{z=(L\otimes P)^{1/2}x} \frac{z^T(L\otimes P)z}{z^Tz} \le \sup\_{z \neq 0}\frac{z^T(L\otimes P)z}{z^Tz} = \|L\otimes P\|.
\end{eqnarray\*}


