Matrix results on the Khatri-Rao and Tracy-Singh products

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Channel Estimation via Orthogonal Matching Pursuit for Hybrid MIMO Systems in Millimeter Wave Communications (2016, background): “For M × N matrices A and B, A ⊗ B denotes the M2 × N2 matrix of Kronecker product between A and B; and A B denotes the M2 ×N matrix of Khatri-Rao product [23] defined as A B = [A (1) ⊗ B (1) , .”
Hadamard, Khatri-Rao, Kronecker and Other Matrix Products (2008, methodology): “For Theorems 5 through 7, see Liu (1999); Albert’s theorem (see e.g. Albert (1969) and Bekker and Neudecker (1989)) plays a role in the proof.”
P-spline ANOVA-type interaction models for spatio-temporal smoothing (2011, methodology): “Using some matrix algebra results (Liu, 1999) it can be shown that the mixed model matrices X and Z calculated as BT , are in fact, the tensor product of the marginal basis: X = X22X1 and (2.”
Inequalities Involving Khatri-Rao Products of Positive Semi-definite Matrices ∗ (2002, background): “i=1 mipi such that Z Z = I and A ∗B = Z (A B)Z (14) for any A ∈ S(m) and B ∈ S(p) which are partitioned as in (1).”