In vector form the disturbances can be written as

where Z_μ = I_N [otimes ] ι_T, I_N is an identity of dimension T, and ι_N is a vector of ones dimension N, Z_λ = ι_N [otimes ] I_T, μ is of dimension N × 1, λ is of dimension T × 1, and ν is of dimension NT × 1. In general, if Δ = [X₁,X₂] then P_Δ = P_X1 + P_{[QX1 X2]}, where P_Δ = Δ(Δ′Δ)⁻¹Δ′ denotes the projection matrix on Δ and Q_Δ = I − P_Δ. Applying this result to Δ = [Z_μ,Z_λ] , one gets

where P = I_N [otimes ] J_T with J_T = ι_Tι_T′/T and Q = I_N [otimes ] E_T with E_T = I_T − J_T. Using the fact that QZ_λ = ι_N [otimes ] E_T, Z_λ′QZ_λ = NE_T, (Z_λ′QZ_λ)⁻ = (1/N)E_T, one gets P_QZλ = J_N [otimes ] E_T. Hence

which means that

as required. Here Q_Δ is the fixed effects transformation derived by Wallace and Hussain (1969). Note that the order does not matter; i.e., one could have orthogonalized on Z_λ.