Richard Hamilton "A matrix Harnack estimate for the heat equation. Communications in Analysis and Geometry. 1(1993), 113-126." On page 125, at the end of the proof of Theorem 4.3, I abstract the question and can be described as (Einstein's summation convention):
Let $P$ be a $(0,4)$-tensor. $P_{ikjl}$ is skew-symmetric in $i$ and $k$, skew-symmetric in $j$ and $l$, symmetric in $\{ik\}$ and $\{jl\}$. For any vectors $v$ and $w$, $P_{ikjl}v_iw_kv_jw_l\ge 0$. Let $(N_{ij})$ be a real symmetric matrix. Prove that if $v$ is a vector satisfying $N_{ij}v_j=0$, then $P_{ikjl}N_{kl}v_iv_j\ge 0$.
How to prove this proposition?
Professor Hamilton says that "it is clear if we diagonalize $N_{kl}$ with respect to a basis". However, I have not thought clearly about this, and the following example seems to negate this proposition.
Suppose that the basic vector space is $2$-dimensional. $N=\begin{bmatrix} -1 & 0\\ 0 & 0 \end{bmatrix}$, $v=\begin{bmatrix} 0\\ 1 \end{bmatrix}$, then $P_{ikjl}N_{kl}v_iv_j=-P_{2121}\le 0$.
What is wrong with this example?