content_image/323448.jpg content_image/323450.jpg content_image/323453.jpg content_image/323449.jpg content_image/323454.jpg
whose column vectors are the two positive P$_{j}$ α . For each I$_{i}$ , we define 

<img src='content_image/323448.jpg'>

which is a ∏ $^{j}$∈I$_{i}$ k$_{j}$ by 2 n$_{i}$ matrix. n$_{i}$ is the number of items in I$_{i}$ . Each P$_{j}$ is a full column rank matrix of rank 2. Thus, P i is rank of 2 n$_{i}$ matrix and is a full column rank matrix. 

Each column vector of T$_{I}$$_{i}$ is a column vector of P $_{i}$. We can show that for two classes α 1 ̸ = α $_{2}$, α $_{1}$-th and α $_{2}$-th columns of T$_{I}$$_{i}$ are not identical. We prove this by contradiction. Suppose that they are the same. It is easy to see that the α $_{l}$-th column in T$_{I}$$_{i}$ has the form 

l = 1 , 2. So 

However, we can find item j ∗ ∈ I$_{i}$ such that 

for all k = 1 , ..., k$_{j}$ ∗ − 1 which means 

<img src='content_image/323450.jpg'>

<img src='content_image/323453.jpg'>

<img src='content_image/323449.jpg'>

<img src='content_image/323454.jpg'>

