In this paper, covariance matrices of heights measured relative to the average height of growing self-affine surfaces in the steady state are investigated in the framework of random matrix theory. We show that the spectral density of the covariance matrix scales as ρ(λ) ~ λ-ν deviating from the prediction of random matrix theory and has a scaling form, ρ(λ, L) = λ-ν f(λ/Lϕ) for the lateral system size L, where the scaling function f(x) approaches a constant for λ ≪ Lϕ and zero for Lϕ≪λ< λ max . The values of exponents obtained by numerical simulations are ν ≈ 1.70 and ϕ ≈ 1.51 for the Edward–Wilkinson class and ν ≈ 1.61 and ϕ ≈ 1.76 for the Kardar–Parisi–Zhang class, respectively. The distribution of the largest eigenvalues follows a scaling form as ρ(λ max , L) = 1/Lb f max ((λ max - La)/Lb), which is different from the Tracy–Widom distribution of random matrix theory while the exponents a and b are given by the same values for the two different classes.