Idempotent matrix algebra

Question

Let $M$ be an $n\times n$ idempotent matrix and $a,b$ are $n\times 1$ vectors with $a\le b\le 0$. Then, $$a^\top Mb \ge 0 ?$$

If so, and if $a\ge b\ge 0$, then the above result still hold true?

Thanks,

Answer 1

Since entrywise non-negativeness of vectors is a basis dependent property, but idempontence is not, the stated conjecture is unlikely true. Counterexample: $$ \pmatrix{-1&-2}\pmatrix{1&-1\\ 0&0}\pmatrix{0\\ -1}=-1<0. $$

Edit. It doesn't help even if you require $M$ to be symmetric. E.g. $$ \pmatrix{-2&-1}\left[\frac1{\sqrt{2}}\pmatrix{1&-1\\ -1&1}\right]\pmatrix{0\\ -1}=-\frac1{\sqrt{2}}<0. $$