A matrix inequality involving pseudoinverse

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.

• Thanks for this information, @AVK. I forgot to mention that W is not a square matrix. Now I've added it into the question statement. – abao5887 Dec 25 '18 at 12:10

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 $$C=AW=\left(\begin{array}{c} 2\\0 \end{array}\right),\quad D=BW=\left(\begin{array}{c} 1\\0 \end{array}\right),$$ $$C^{+}=\left(\begin{array}{cc} \frac12 & 0 \end{array}\right),\quad D^{+}=\left(\begin{array}{cc} 1&0 \end{array}\right),$$ $$CC^{+}=DD^{+}=\left(\begin{array}{cc} 1&0\\0&0 \end{array}\right).$$
• Thanks for this example. Actually, I found that there was something wrong in the previous question, the condition should be $B \succ A$. Afterall, this example still illustrates the problem. Furthermore, I might trace back to the proof of a more general question, which is the origin of the stated one, see (math.stackexchange.com/questions/3052118/…). – abao5887 Dec 25 '18 at 13:52