Is it true that if I have an $(n \times n)$ positive definite matrix $M$ and two $(m \times 1)$ vectors $a$ and $b$, that:
$$a\geq_e b \implies Ma \geq_e Mb$$
Where $\geq_e$ represents elementwise maximum (i.e. if every element of $a$ is greater than or equal to it's corresponding element in $b$, the $a \geq_e b$). If so why? If not does anyone have a counter example?
The answer is no. For example, $$ M = \pmatrix{2&-1\\-1&2}, \quad a = \pmatrix{1\\0}, b = \pmatrix{0\\0}. $$