Is $z_1^T z_1 A + z_2^T z_2 A ... + z_n^T z_n A = Z^T Z A$ where the $z_1, \cdots ,z_n$ are the row vectors of the matrix $Z$ for all $n$?
The particular term appears as part of the whitening equation in the Equivariant adaptive source separation paper. Also a similar term also appears in Efficient covariance matrix update for variable metric evolution strategies. It is also related to square root filtering. Knowing that the above term holds means I could adapt the algorithm from the EASS paper for batch learning. The question itself something that could easily be verified to be true or false with a simple loop through the Z matrix and an accumulating state, but it is easier to just ask here.
If $z_k$ denotes the $k$th row of the matrix $Z$, then we have $$ z_1^Tz_1A + \cdots + z_n^Tz_nA = \\ (z_1^Tz_1 + \cdots + z_n^Tz_n)A = \\ \pmatrix{z_1^T & z_2^T & \cdots & z_n ^T} \pmatrix{z_1\\z_2\\ \vdots \\ z_n}A =\\ Z^TZA $$