I have a matrix inequality that I think is true, but I can't prove.
$D_1$ and $D_2$ are diagonal matrices with non-negative entries. $M_1$ and $M_2$ are positive definite matrices.
I want to show $\|D_1 M_1 + D_2 M_2\|_2 \leq \displaystyle\max_{i=1,2}\|D_i\|_2 \|M_1 + M_2\|_2$.
Using the triangle inequality, I can show $\|D_1 M_1 + D_2 M_2\|_2 \leq \displaystyle\max_{i=1,2}\|D_i\|_2 \left( \|M_1\|_2 + \|M_2\|_2\right)$, but I want $M_1$ and $M_2$ to stay together inside the norm.
Any ideas are appreciated.
The inequality does not hold in general. Counterexample: $$ M_1=\pmatrix{2&-1\\ -1&2},\ M_2=\pmatrix{2&1\\ 1&2},\ D_1=\pmatrix{1&0\\ 0&0},\ D_2=I. $$ We have $\|D_1M_1+D_2M_2\|_2=\left\|\pmatrix{4&0\\ 1&2}\right\|_2\approx4.16> \max\limits_{i=1,2}\|D_i\|_2 \|M_1 + M_2\|_2=4$.