A matrix inequality involving pseudoinverse

Question

I am trying to solve a problem in the context of signal processing, which leads to the following question. Consider two positive definite Hermitian matrices $A$ and $B$, and a full cloumn rank matrix $W$ with more rows than columns. Assume $B \succ A$, which means $B-A$ is positive definite. Let $C=AW$, and $D=BW$. I want to prove the following inequality $$CC^+ \succ DD^+.$$ Here $C^+$ refers to $C^+=(C^HC)^{-1}C^H$. However, I didn't find any good approach to this proof. Is this inequality true? In addition, I also wonder if other necessary conditions on the involving matrices are needed.

Answer 1

Consider the case $$ A=\left(\begin{array}{cc} 2&0\\0&2 \end{array}\right),\quad B=\left(\begin{array}{cc} 1&0\\0&1 \end{array}\right),\quad W=\left(\begin{array}{c} 1\\0 \end{array}\right). $$ We have $$ C=AW=\left(\begin{array}{c} 2\\0 \end{array}\right),\quad D=BW=\left(\begin{array}{c} 1\\0 \end{array}\right), $$ $$ C^{+}=\left(\begin{array}{cc} \frac12 & 0 \end{array}\right),\quad D^{+}=\left(\begin{array}{cc} 1&0 \end{array}\right), $$ $$ CC^{+}=DD^{+}=\left(\begin{array}{cc} 1&0\\0&0 \end{array}\right). $$