Web - based Supplementary Materials for “ Comparing Large Co - variance Matrices under Weak Conditions on the Dependence Structure and its Application to Gene Clustering ” , by Jinyuan

First we introduce some of the basic notations. For any vector u = (u1, . . . , up) T ∈ R, denote by |u|q the vector `q-norm defined by |u|q = (∑p k=1 |uk| )1/q for q ≥ 1 and write |u|0 = ∑p k=1 I(uk 6= 0). For any set S, denote by S its complement. For a matrix A = (ak`) ∈ Rp×p, we denote by ‖A‖2 the spectral norm, ‖A‖F the Frobenius norm, and ‖A‖1 = ∑p k,`=1 |ak`| the elementwise `1-norm. Recall that N = n + m is the total sample size, p̄ = p(p+ 1)/2 denotes the ambient dimension and the index pairs {(k, `)}1≤k≤`≤p are organized as follows: