Monotonicity of quadratic forms in positive semidefinite Matrices

Question

I'm interested in some kind of monotonicity quality of quadratic forms. Let $A$ be a symmetric positive semidefinite (n x n)-matrix. Suppose we have two real n-dimensional vectors a and b, for which holds $0 \leq a \leq b$, where "$\leq$" is applied componentwise. Does \begin{align} a'Aa \leq b'Ab \end{align} now also hold? For diagonal matrix this holds, since it can only have positive entries, due to the positive semidefiniteness.
My general proof idea is that $Da=b$ for $D=diag(b_1/a_1,\dots, b_n/a_n)$ which leads to $b'Ab - a'Aa = a'(DAD-A)a$. Now if $DAD-A$ was positive semidefinite, this would conclude the proof. This where I'm stuck.

Answer 1

take $$ A = \left( \begin{array}{rr} 1 & -5 \\ -5 & 26 \\ \end{array} \right) $$

with $$ a = \left( \begin{array}{r} 1 \\ 1 \\ \end{array} \right) $$

$$ b = \left( \begin{array}{r} 15 \\ 3 \\ \end{array} \right) $$