T$_{˜}$ $_{I$_{3}$}$. Therefore, T$_{˜}$ $_{I$_{1}$}$,T$_{˜}$ I$_{2}$ and T$_{˜}$ I$_{3}$ are all full column rank. We define where Λ is a M by M diagonal matrix with π α being its α -th element. It is not hard to see that every entry of array W corresponds to a probability P ( Y 1 = y $^{1}$,... ,Y J = y $^{J}$). Suppose that there is another decomposition of W say W = [ T ′ ˜ $_{I$_{1}$}$Λ $^{′}$,T ′ ˜ $_{I$_{2}$}$,T ′ ˜ $_{I$_{3}$}$]. Notice that each T$_{˜}$ I$_{i}$ has rank M and M + M + M ≥ 2 M + 2 provided M ≥ 2. Then we apply Lemma 1 and obtain that T$_{˜}$ $_{I$_{1}$}$Λ = T ′ ˜ $_{I$_{1}$}$Λ $^{′}$PA , T$_{˜}$ I$_{2}$ = T ′ ˜ $_{I$_{2}$}$PB , and T$_{˜}$ I$_{3}$ = T ′ ˜ $_{I$_{3}$}$PC . Here, A,B , and C are all diagonal matrix with ABC = I and I is an identity and P is a column permutation matrix. Each column of T$_{˜}$ I$_{i}$ and T ′ ˜ I$_{i}$ corresponds to a probability distribution and thus sums up to one. It means A,B and C must be identity matrix. Hence, we conclude that T$_{˜}$ $_{I$_{1}$}$Λ = T ′ ˜ $_{I$_{1}$}$Λ $^{′}$P which implies Λ = P $^{′}$Λ$^{′}$P . Then, we have T$_{˜}$ I$_{1}$ = T ′ ˜ $_{I$_{1}$}$P , T$_{˜}$ I$_{2}$ = T ′ ˜ $_{I$_{2}$}$P and T$_{˜}$ I$_{3}$ = T ′ ˜ $_{I$_{3}$}$P . This is equivalent that the item parameters p k j α and the latent class population π α are identifiable up to a permutation of the class label. Proof of Corollary 1. Without loss of generality, we assume that the first, second, and third K rows of Q each form an identity matrix. The attributes are binary and each of the first 3 K items only depends on one attribute. Thus, their item response function p$_{j}$ α can only take two possible values. Furthermore, we divide these 3 K items into 3 groups I$_{1}$ = { 1 ,... ,K } , I$_{2}$ = { K +1 , · · · , 2 K } , and I$_{3}$ = { 2 K + 1 , · · · , 3 K } . It is straightforward to check that these three subsets of items satisfy condition A1 in Theorem 1. The corollary is an application of Theorem 1. Proof of Theorem 2. Under condition B 1, we define W = [ T$_{˜}$ $_{I$_{1}$}$Λ ,T$_{˜}$ $_{I$_{2}$}$,T$_{˜}$ $_{I$_{3}$}$] , P$_{j}$ =     p 1 $^{j}$− p 1 j + p 2 $^{j}$− p 2 j + . . p k$_{j}$ $^{j}$− p k$_{j}$ j +    $^{,}$ 36