we have $$M(k^{\prime },q_{0}^{\prime })=A(k,q_{0})M(k,q_{0})A^{-1}(k,q_{0})
\label{eq:10}$$and the sufficient and necessary condition for the projector on noncommutative orbifold $T^{2}/Z_{N}$ to satisfy is: $$\begin{aligned}
M(k,q_{0})^{2} &=&M(k,q_{0}), \\
M(k^{\prime },q_{0}^{\prime })
&=&A(k,q_{0})M(k,q_{0})A^{-1}(k,q_{0}).\label{eq:24}\end{aligned}$$Next we will study the relation between coefficient function $\psi
_{st}(k,q) $ and the reduced matrix $M(k,q_{0})$. From (\[eq:16\])(\[eq:21\])(\[eq:22\]) and (\[eq:23\])we have $$\begin{aligned}
P|k,q_{0}+\frac{ln}{A}>= &&\sum_{s,t}u_{1}^{s}u_{2}^{t}\Psi
_{st}(u_{1}^{A},u_{2}^{A})|k,q_{0}+\frac{ln}{A}>  \nonumber \\
&=&\sum_{s,t}e^{-2\pi i(q_{0}/l+n/A)t}\psi _{st}(k,q_{0})|k,q_{0}+\frac{%
l(n+s)}{A}>  \nonumber \\
&=&\sum_{n^{\prime }}M(k,q_{0})_{n^{\prime }n}|k,q_{0}+\frac{ln^{\prime }%
}{A}>.\end{aligned}$$So for $n+s<A$ case, we have $$M(k,q_{0})_{n+s,n}=\sum_{t=0}^{A-1}e^{-2\pi i(q_{0}/l+n/A)t}\psi
_{st}(k,q_{0})$$and for $n+s\geq A$ case, we have $$M(k,q_{0})_{n+s-A,n}=\sum_{t=0}^{A-1}e^{-2\pi i(q_{0}/l+n/A)t}\psi
_{st}(k,q_{0})e^{-ilk}.$$Setting $$M(k,q_{0})_{n+s,n}=M(k,q_{0})_{n+s-A,n}e^{ilk},  \label{eq:1}$$We can uniformly write as: $$M(k,q_{0})_{n+s,n}=\sum_{t=0}^{A-1}e^{-2\pi i(q_{0}/l+n/A)t}\psi
_{st}(k,q_{0})  \label{eq:2}$$and have $$\psi
_{st}(k,q_{0})=\frac{1}{A}\sum_{n=0}^{A-1}M(k,q_{0})_{n+s,n}e^{2\pi
i(q_{0}/l+n/A)t}.  \label{eq:3}$$Eq.(\[eq:2\]) and (\[eq:3\]) is the relation between $\psi
_{st}$ and the elements of reduced matrix $M$.

Continuous solution for the Projector with Minimal Trace
========================================================

Now one may ask what property the reduced matrix $M$ possess when the coefficient function $\psi _{st}$ is a continuous function. In this section, we mainly answer this question. First we prove the $A\times A$ matrix satisfying the condition $M^{2}=M$ is always diagonalizable. For any vector $%
\psi $, $M\psi $ is invariant under $M$, namely $$M(M\psi )=M\psi .$$Assume there are totally $B$ linear independent invariant vectors under transformation $M$, then(1) for $A=B$ case, the matrix $M$ 