Let X be an n × p matrix, Z be an n × q matrix, β be a p-vector, δ be a q-vector. Let X_{k} denote the kth row of X and likewise for Z_{k}.

Assuming that Z is not all zeros, is it true that

∑_{i=1}^{n} ∑_{j=1}^{n} |X_{i}β + Z_{i}δ − X_{j}β − Z_{j}δ| = ∑_{i=1}^{n} ∑_{j=1}^{n} |X_{i}β − X_{j}β|  (1)

if and only if

δ ≡ 0  (2)

If it is not true in general, under what conditions is it true? Note that in a trivial case |a + b| = |a| can be true if b = 0 or b = −2a.