Let A be an $a\times b$ real matrix where $ b \leq a$. The $b \times b$ matrix $D$ is a real symmetric positive semidefinite matrix. The matrix $D$ may not be invertible. Then under what condition $trace(A'A)$ can be less than $trace(A'DA)$? (Or the conditions that lead $trace(A'A)$ to be smaller than $trace(A'D'DA)$ are okay too.)
Any reference, comments or answers are appreciated.
PS. There's no restriction on the eigenvalues of $D$, other than it's real and not negative. Some eigenvalues of D can be zero, or very close to zero, or smaller than one. But the sum of eigenvalues of D (or the trace of D equivalently) is larger than $a$, which is the sum of eigenvalues (or trace) of $I_a$.