Zero entries in multipartite product unitary matrices

Abstract

Characterizing the zero entries in multipartite unitary matrices plays an important role in evaluating the usefulness of such matrices in quantum information. We investigate the zero entries in unitary and product unitary matrices. On one hand, we study the quantity properties of the zero entries in the multiqubit and bipartite product unitary matrices, respectively. We also investigate the ratio of the number of elements in the set made up of the numbers of zero entries in product unitary matrices to the number of all cases of such entries. On the other hand, we focus on the distribution of zero entries in unitary and product unitary matrices. Our results help identify whether a matrix containing at least one zero entry is a unitary matrix.

Appendices

Appendix A: The proof of Theorem 12

Proof

• (i) If $U_j=I_{k_j}$ and $j\in [1,m]$, then $U=\otimes ^m_{j=1}U_j=I_{2^n}$. So we have $4^n-2^n\in {{{\mathcal {P}}}}_n(2^{k_1},2^{k_2},\ldots ,2^{k_m})|_{{\mathbb {U}}^{2^n} \cap {\mathbb {P}}^m}$. This is the case where we obtain the maximum of ${{{\mathcal {P}}}}_n(2^{k_1},2^{k_2},\ldots ,2^{k_m})|_{{\mathbb {U}}^{2^n} \cap {\mathbb {P}}^m}$. If $U_j$ for $j\in [1,m]$ is a $2^{k_j}\times 2^{k_j}$ unitary matrix with no zero entries, then $U=\otimes ^m_{j=1}U_j$ has no zero entries. So we have $0\in {{{\mathcal {P}}}}_n(2^{k_1},2^{k_2},\ldots ,2^{k_m})|_{{\mathbb {U}}^{2^n} \cap {\mathbb {P}}^m}$. This is the case where we obtain the minimum of ${{{\mathcal {P}}}}_n(2^{k_1},2^{k_2},\ldots ,2^{k_m})|_{{\mathbb {U}}^{2^n} \cap {\mathbb {P}}^m}$.

• (ii) Using (5), we have $z(U)=4^n-\prod ^m_{j=1}(4^{k_j}-z(U_{2^{k_j}}))$. Suppose $m<n$, i.e., $k_{\textit{max}}=\textit{max}\{k_1,k_2,\ldots ,k_m\}>1$. If $z(U_{2^{k_{\textit{max}}}})=1$ and $z(U_{2^{k_j}})=0$ for $k_j \ne k_{\textit{max}}$, then we can obtain that $4^n-4^{n-k_{\textit{max}}}(4^{k_{\textit{max}}}-1)\in {{{\mathcal {P}}}}_n(2^{k_1},2^{k_2},\ldots ,2^{k_m})|_{{\mathbb {U}}^{2^n} \cap {\mathbb {P}}^m}$. This is the case where we obtain the nonzero minimum of ${{{\mathcal {P}}}}_n(2^{k_1},2^{k_2},\ldots ,2^{k_m})|_{{\mathbb {U}}^{2^n} \cap {\mathbb {P}}^m}$.

• (iii) Suppose $m=n$. Then, U is an n-qubit product unitary. Using (5), we have $z(U)=4^n(1-2^{-k})$. If $k=1$, then we can obtain that $2^{2n-1}\in {{{\mathcal {P}}}}_n(2^{k_1},2^{k_2},\ldots ,2^{k_m})|_{{\mathbb {U}}^{2^n} \cap {\mathbb {P}}^m}$. This is the case where we obtain the nonzero minimum.

• (iv) Using (5), we have $z(U)=4^n-\prod ^m_{j=1}(4^{k_j}-z(U_{2^{k_j}}))$. If we permute $\{k_1,k_2,\ldots ,k_m\}$ and obtain $\{k_1',k_2',\ldots ,k_m'\}$, then one can verify that $z(U')=4^n-\prod ^m_{j'=1}(4^{k_j'}-z(U_{2^{k_j'}}))=4^n-\prod ^m_{j=1}(4^{k_j}-z(U_{2^{k_j}}))=z(U)$.

• (v) Using (5), we have $z(U)=4^n-\prod ^m_{j=1}(4^{k_j}-z(U_{2^{k_j}}))$. If $k_{\textit{min}}=1$, then the value of $4-z(U_{k_{\textit{min}}})$ is 2 or 4. Then, $4^n$ and $\prod ^m_{j=1}(4^{k_j}-z(U_{2^{k_j}})$ are obviously even numbers. So z(U) is an even number, and ${{{\mathcal {P}}}}_n(2^{k_1},2^{k_2},\ldots ,2^{k_m})|_{{\mathbb {U}}^{2^n} \cap {\mathbb {P}}^m}$ is a set of even numbers.

• (vi) Using (5), we have $z(U)=4^n-\prod ^m_{j=1}(4^{k_j}-z(U_{2^{k_j}}))$. If $z(U_j) \in \zeta _{2^{k_j}}$ and $z(U_i)=0$ for $i\ne j$, then we have

  $$\begin{aligned} z(U)&=4^n-4^{n-k_j}(4^{k_j}-z(U_{2^{k_j}})) \end{aligned}$$

  (A1)
  $$\begin{aligned}&\quad =4^{n-k_j}(4^{k_j}-4^{k_j}+z(U_{2^{k_j}})) \end{aligned}$$

  (A2)
  $$\begin{aligned}&\quad =4^{n-k_j}z(U_{2^{k_j}}). \end{aligned}$$

  (A3)

  So we obtain that $4^{n-k_j}\zeta _{2^{k_j}} \subseteq {{{\mathcal {P}}}}_n(2^{k_1},2^{k_2},\ldots ,2^{k_m})|_{{\mathbb {U}}^{2^n} \cap {\mathbb {P}}^m}$.

• (vii) It follows from [21, Theorem 2.5] that for all $k_j$, the value of $z(U_{2^{k_j}})$ can be $4^{k_j}\frac{d_j}{2^{k_j}}$, and $d_j\in [0,2^{k_j}-1]$. Then, we have

  $$\begin{aligned} z(U)=4^n-\prod ^m_{j=1}(4^{k_j}-4^{k_j}\frac{d_j}{2^{k_j}})=4^n-2^n\prod ^m_{j=1}(2^{k_j}-d_j), \end{aligned}$$

  (A4)

  where $d_j\in [0,2^{k_j}-1]$. Obviously, $(2^{k_j}-d_j)\in [1,2^{k_j}]$. We can obtain that if $b_1,b_2,\ldots ,b_m\in {\mathbb {N}}$ satisfy $b_1 \le 2^{k_1}, b_2 \le 2^{k_2}, \ldots ,b_m \le 2^{k_m}$, then $4^n-2^nb_1b_2\ldots b_m \in {{{\mathcal {P}}}}_n(2^{k_1},2^{k_2},\ldots ,2^{k_m})$.

• (viii) One can verify the claim by the expression $z(U)=4^n-\prod ^m_{j=1}(4^{k_j}-z(U_{2^{k_j}}))$ in (5).$\square$

Appendix B: The proof of Theorem 14

Proof

• (i) If $U_{m_1}=I_{m_1}$, $U_{m_2}=I_{m_2}$, then $U=U_{m_1}\otimes U_{m_2}=I_{m_1m_2}$. So we have $m_1^2m_2^2-m_1m_2\in {{{\mathcal {P}}}}(m_1,m_2)|_{{\mathbb {U}}^{m_1}\times {\mathbb {U}}^{m_2}}$. This is the case where we obtain the maximum of ${{{\mathcal {P}}}}(m_1,m_2)|_{{\mathbb {U}}^{m_1}\times {\mathbb {U}}^{m_2}}$. If $U_{m_i}$ for $i=1,2$ is a $2^{m_i}\times 2^{m_i}$ unitary matrix with no zero entries, then $U=U_{m_1}\otimes U_{m_2}$ has no zero entries. So we have $0\in {{{\mathcal {P}}}}(m_1,m_2)|_{{\mathbb {U}}^{m_1}\times {\mathbb {U}}^{m_2}}$. This is the case where we obtain the minimum of ${{{\mathcal {P}}}}(m_1,m_2)|_{{\mathbb {U}}^{m_1}\times {\mathbb {U}}^{m_2}}$.

• (ii) Using (6), we have $z(U)=m_1^2 m_2^2-(m_1^2-z(U_{m_1}))(m_2^2-z(U_{m_2}))$. Suppose $m_{\textit{max}}=\textit{max}\{m_1,m_2\}>2$. If $z(U_{m_{\textit{max}}})=1$ and $z(U_{m_j})=0$ for $j\ne {\textit{max}}$, then we can obtain that $m_1^2m_2^2-(m_1+m_2-m_{\textit{max}})^2({m_{\textit{max}}^2}-1)\in {{{\mathcal {P}}}}(m_1,m_2)|_{{\mathbb {U}}^{m_1}\times {\mathbb {U}}^{m_2}}$. This is the case where we obtain the nonzero minimum of ${{{\mathcal {P}}}}(m_1,m_2)|_{{\mathbb {U}}^{m_1}\times {\mathbb {U}}^{m_2}}$.

• (iii) Using (6), we have ${{{\mathcal {P}}}}(m_1,m_2)|_{{\mathbb {U}}^{m_1}\times {\mathbb {U}}^{m_2}}=\{z(u)|z(U)=m_1^2 m_2^2-(m_1^2-z(U_{m_1}))(m_2^2-z(U_{m_2}))\}={{{\mathcal {P}}}}(m_2,m_1)|_{{\mathbb {U}}^{m_2}\times {\mathbb {U}}^{m_1}}$.

• (iv) If $m_1=2$, then using (6) we have ${{{\mathcal {P}}}}(2,m_2)|_{{\mathbb {U}}^2\times {\mathbb {U}}^{m_2}}=\{z(U)|z(U)=4m_2^2-(4-z(U_2))(m_2^2-z(U_{m_2}))\}$. We review [21, Theorem 2.5] that the value of $z(U_2)$ is 0 or 2. On one hand, if $z(U_2)=0$, then $z(U)=4\zeta _{m_2}$. On the other hand, if $z(U_2)=2$, then $z(U)=2m^2_2-2\zeta _{m_2}$. So ${{{\mathcal {P}}}}(2,m_2)|_{{\mathbb {U}}^2\times {\mathbb {U}}^{m_2}}=4\zeta _{m_2}\cup (2m^2_2-2\zeta _{m_2})$. Obviously, z(U) is an even number and ${{{\mathcal {P}}}}(2,m_2)|_{{\mathbb {U}}^2\times {\mathbb {U}}^{m_2}}$ is an set of even numbers. Similarly, if $m_2=2$, then ${{{\mathcal {P}}}}(2,m_2)|_{{\mathbb {U}}^2\times {\mathbb {U}}^{m_2}}$ is also a set of even numbers.

• (v) Using (6), we have $z(U)=m_1^2 m_2^2-(m_1^2-z(U_{m_1}))(m_2^2-z(U_{m_2}))$. If $z(U_{m_1})=0$, then we have $z(U)=m_1^2z(U_{m_2})$. So $m^2_1\zeta _{m_2}\subseteq {{{\mathcal {P}}}}(m_1,m_2)|_{{\mathbb {U}}^{m_1}\times {\mathbb {U}}^{m_2}}$. Similarly, $m^2_2\zeta _{m_1}\subseteq {{{\mathcal {P}}}}(m_1,m_2)|_{{\mathbb {U}}^{m_1}\times {\mathbb {U}}^{m_2}}$.

• (vi) Suppose $m_1,m_2\ne 3$. It follows from [21, Theorem 2.5] that for $m_i$, $z(U_{m_1})$ can be $m_i^2\frac{d_i}{m_i}$, and $d_i\in [0,m_i-1]$. So using (6), we have

  $$\begin{aligned} z(U)= & {} m_1^2 m_2^2-(m_1^2-m_1^2\frac{d_1}{m_1})(m_2^2-m_2^2\frac{d_2}{m_2})\nonumber \\= & {} m_1m_2(m_1m_2-(m_1-d_1)(m_2-d_2)), \end{aligned}$$

  (B1)

  where $m_1-d_1\in [1,m_1]$, and $m_2-d_2\in [1,m_2]$. We can obtain that if $b_1,b_2 \in {\mathbb {N}}$ satisfy $b_1 \le m_1, b_2 \le m_2$, then $m_1^2m_2^2-m_1m_2b_1b_2 \in {{{\mathcal {P}}}}(m_1,m_2)|_{{\mathbb {U}}^{m_1}\times {\mathbb {U}}^{m_2}}$.

• (vii) One can verify the claim by the expression $z(U)=m_1^2 m_2^2-(m_1^2-z(U_{m_1}))(m_2^2-z(U_{m_2}))$ in (6).$\square$

Appendix C: The proof of Theorem 16

Proof

• (i) Suppose the number of elements in $\zeta _{2^{k_j}}$ for $j\in [1,m]$ is $N_j$, respectively. It follows from [21, Theorem 2.5] that for all $k_j$, we have $2^{2k_j-1}\le N_j<4^{k_j}-2^{k_j}$. Using (5), we have $z(U)=4^n-\prod ^m_{j=1}(4^{k_j}-z(U_{2^{k_j}}))$. Then, we can obtain that

  $$\begin{aligned} N<N_1N_2\ldots N_m\le (4^{k_1}-2^{k_1})(4^{k_2}-2^{k_2})\ldots (4^{k_m}-2^{k_m}). \end{aligned}$$

  (C1)

  In addition, we have $N>N_j\ge 2^{2k_j-1}$ for $j \in [1,m]$. So we obtain

  $$\begin{aligned} N> 2^{2k_{\textit{max}}-1}, \end{aligned}$$

  (C2)

  and $k_{\textit{max}}=\textit{max}\{k_1, k_2,\ldots ,k_m\}$.

• (ii) Suppose $m=n$, i.e., $k_1=k_2=\ldots =k_m=1$. That is n-qubit product unitary matrix in Fig. 1. Using (4), we have $N=n+1$. Then, we obtain that the ratio

  $$\begin{aligned} \frac{N}{4^n}=\frac{n+1}{4^n} \end{aligned}$$

  (C3)

  Obviously, the ratio approaches zero when n approaches the infinity.

• (iii) Suppose $\frac{n}{m}$ is a finite number and $k_1=k_2=\ldots =k_m$. That is to say that the dimension of every partite of the matrix U is a finite number $\frac{n}{m}$. Using (C1), we obtain that the ratio

  $$\begin{aligned} \frac{N}{4^n} < \frac{N_1N_2\ldots N_m}{4^n} \le \frac{(4^{\frac{n}{m}}-2^{\frac{n}{m}})^m}{4^n}=(1-\frac{1}{2^{\frac{n}{m}}})^m . \end{aligned}$$

  (C4)

  We have $0<1-\frac{1}{2^{\frac{n}{m}}}<1$. Since $\frac{n}{m}$ is a finite number, m approaches the infinity when n approaches the infinity. Then, we have $(1-{2^{-\frac{n}{m}}})^m$ approaches zero when n approaches the infinity. Then, we obtain that the ratio $N/4^n$ approaches zero when n approaches the infinity.

• (iv) Suppose $k_1=k_2=\ldots =k_m$. That is to say that the dimensions of $U_1, U_2,\ldots , U_m$ are all $\frac{n}{m}$. On one hand, using (C1) we obtain that the ratio

  $$\begin{aligned} \frac{N}{4^n} < \frac{N_1N_2\ldots N_m}{4^n} \le \frac{{(4^{\frac{n}{m}}-2^{\frac{n}{m}})^m}}{{4^n}}=(1-\frac{1}{2^{\frac{n}{m}}})^m . \end{aligned}$$

  (C5)

  Because m is a finite number, we obtain that $(1-\frac{1}{2^{\frac{n}{m}}})^m$ approaches one when n approaches the infinity.

  On the other hand, using (C2) we obtain that the ratio

  $$\begin{aligned} \frac{N}{4^n} > \frac{N_1}{4^n}\ge \frac{2^{\frac{2n}{m}-1}}{{4^n}}=\frac{1}{2} \frac{1}{4^{n(1-\frac{1}{m})}} . \end{aligned}$$

  (C6)

  Because m is a finite number, we obtain that $\frac{1}{4^{n(1-\frac{1}{m})}}$ approaches zero when n approaches the infinity.

• (v) Suppose $k_2=\ldots =k_m=1$. That is to say when n increases to approach infinity, only $k_1$ increases to approach the infinity and $k_2,\ldots ,k_m$ are invariable. On one hand, using (C1) we obtain that the ratio

  $$\begin{aligned} \frac{N}{4^n} < \frac{N_1N_2\ldots N_m}{4^n} \le \frac{(4^{n-m+1}-2^{n-m+1})2^{m-1}}{{4^n}}=\frac{4^{n-m+1}-2^{n-m+1}}{4^{n-m+1}} 2^{1-m}.\nonumber \\ \end{aligned}$$

  (C7)

  If m is a finite number, then we have $\frac{4^{n-m+1}-2^{n-m+1}}{4^{n-m+1}}$ approaches one when n approaches the infinity. So we obtain the upper bound of the ratio $\frac{N}{4^n}$ is $2^{1-m}$ when n approaches the infinity.

  On the other hand, using (C2) we obtain that the ratio

  $$\begin{aligned} \frac{N}{4^n} > \frac{N_1}{4^n}\ge \frac{2^{2(n-m+1)-1}}{{4^n}}= 4^{-m-1} . \end{aligned}$$

  (C8)

  If m is a finite number, then we obtain the lower bound of the ratio $\frac{N}{4^n}$ is $4^{-m-1}$ when n approaches the infinity.

$\square$

Appendix D: The proof of Theorem 18

Proof

• (i) Suppose the numbers of integers in $\zeta _{m_1}$ and $\zeta _{m_2}$ are $M_1$ and $M_2$, respectively. It follows from [21, Theorem 2.5] that $\frac{1}{2}m_1^2\le M_1<(m_1^2-m_1)$ and $\frac{1}{2}m_2^2\le M_2<(m_2^2-m_2)$. Using (6), we obtain that

  $$\begin{aligned} M<M_1M_2<(m_1^2-m_1)(m_2^2-m_2). \end{aligned}$$

  (D1)

  In addition, we have $M>M_1\ge \frac{1}{2}m_1^2$ and $M>M_2\ge \frac{1}{2}m_2^2$. So we obtain that

  $$\begin{aligned} M>\frac{1}{2}m_{\textit{max}}^2, \end{aligned}$$

  (D2)

  and $m_{\textit{max}}=\textit{max}\{ m_1, m_2\}$.

• (ii) Suppose $m_1=m_2$. That is to say that when n increases to approach infinity, $m_1$ and $m_2$ increase to approach infinity at a same speed. On one hand, using (D1) we obtain

  $$\begin{aligned} \frac{M}{n^2}<\frac{M_1M_2}{n^2}<\frac{(n-n^{\frac{1}{2}})^2}{n^2}=(1-\frac{1}{n^{\frac{1}{2}}})^2. \end{aligned}$$

  (D3)

  We have $\frac{M}{n^2}<(1-\frac{1}{n^{\frac{1}{2}}})^2$. On the other hand, using (D2) we obtain

  $$\begin{aligned} \frac{M}{n^2}>\frac{M_1}{n^2}>\frac{\frac{1}{2}n}{n^2}=\frac{1}{2n}. \end{aligned}$$

  (D4)

  So we have $\frac{M}{n^2}>\frac{1}{2n}$.

• (iii) Suppose $m_2$ is finite number. That is to say that when n increases to approach the infinity, only $m_1$ increase to approach the infinity. On one hand, using (D1) we obtain that the ratio

  $$\begin{aligned} \frac{M}{n^2}<\frac{M_1M_2}{n^2}<\frac{(\frac{n}{m_2})^2-\frac{n}{m_2}}{(\frac{n}{m_2})^2}\frac{m_2^2-m_2}{m_2^2}=\frac{n-m_2}{n}\frac{m_2-1}{m_2} \end{aligned}$$

  (D5)

  We have $\frac{n-m_2}{n}$ approaches 1 when n approaches the infinity. Then, we obtain that the upper bound of the ratio $\frac{M}{n^2}$ is $\frac{h-1}{h}$. On the other hand, using (D2) we obtain that the ratio

  $$\begin{aligned} \frac{M}{n^2}>\frac{M_1}{n^2}>\frac{(\frac{1}{2}\frac{n}{m_2})^2}{(\frac{n}{m_2})^2}\frac{1}{m_2^2}=\frac{1}{2m_2^2}. \end{aligned}$$

  (D6)

  We obtain that the lower bound of the ratio $\frac{M}{n^2}$ is $\frac{1}{2m_2^2}$.

$\square$

Appendix E: The proof of Lemma 24

Proof

Suppose U is a unitary matrix in ${\mathbb {U}}^3$. Because U has full rank. Each column or each row has no more than two zero entries.

• (i) Suppose the number of zero entries in U is one. One can verify that there is only one ZED-equivalent class, i.e., $\begin{bmatrix} 0 &{} * &{} *\\ * &{} * &{} *\\ * &{} * &{} * \end{bmatrix}.$

• (ii) Suppose the number of zero entries in U is four. If three columns and three rows have zero entries, then there are two ZED-equivalent classes, i.e., $\begin{bmatrix} * &{} 0 &{} 0\\ 0 &{} * &{} *\\ 0 &{} * &{} * \end{bmatrix}$ and $\begin{bmatrix} * &{} 0 &{} 0\\ 0 &{} * &{} *\\ * &{} 0 &{} * \end{bmatrix}$. Obviously, the second class of matrices are not unitary matrices. If three columns and two rows have zero entries, then there is one ZED-equivalent classes, i.e., $\begin{bmatrix} * &{} 0 &{} 0\\ 0 &{} * &{} 0\\ * &{} * &{} * \end{bmatrix}$. Obviously, this class of matrices is not unitary matrices. If two columns and two rows have zero entries, then there is one ZED-equivalent class, i.e., $\begin{bmatrix} * &{} 0 &{} 0\\ * &{} 0 &{} 0\\ * &{} * &{} * \end{bmatrix}$. Obviously, this class of matrices is not unitary matrices. So there is only one ZED-equivalent class, i.e., $\begin{bmatrix} * &{} 0 &{} 0\\ 0 &{} * &{} *\\ 0 &{} * &{} * \end{bmatrix}$.

• (iii) Suppose the number of zero entries in U is six. Because U has full rank, there is only one ZED-equivalent class, i.e., $\begin{bmatrix} * &{} 0 &{} 0\\ 0 &{} * &{} 0\\ 0 &{} 0 &{} * \end{bmatrix}$.

$\square$

Appendix F: The proof of Lemma 25

Proof

Suppose U is a unitary matrix in ${\mathbb {U}}^4$. Because U has full rank. When the number of zero entries of U is under six, each column or each row has no more than three zero entries.

• (i) Suppose the number of zero entries in U is one. One can verify that there is only one ZED-equivalent class, i.e., $\begin{bmatrix} 0 &{} * &{} * &{} *\\ * &{} * &{} * &{} *\\ * &{} * &{} * &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$.

• (ii) Suppose the number of zero entries in U is two. There are two zero entries distribution-equivalent classes, i.e., $\begin{bmatrix} 0 &{} * &{} * &{} *\\ * &{} 0 &{} * &{} *\\ * &{} * &{} * &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$ and $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ * &{} * &{} * &{} *\\ * &{} * &{} * &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$. One can verify that these two classes of matrices can be unitary matrices.

• (iii) Suppose the number of zero entries in U is three. If three columns and three rows have zero entries, then there is one ZED-equivalent class, i.e., $\begin{bmatrix} 0 &{} * &{} * &{} *\\ * &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$.

  If three columns and two rows have zero entries, then there is one ZED-equivalent class, i.e., $\begin{bmatrix} 0 &{} * &{} 0 &{} *\\ * &{} 0 &{} * &{} *\\ * &{} * &{} * &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$.

  Obviously, this class of matrices cannot be unitary matrices.

  If two columns and two rows have zero entries, then there is one ZED-equivalent class, i.e., $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ * &{} 0 &{} * &{} *\\ * &{} * &{} * &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$.

  One can verify this class of matrices can be unitary matrices. So we have two ZED-equivalent classes, $\begin{bmatrix} 0 &{} * &{} * &{} *\\ * &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$ and $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ * &{} 0 &{} * &{} *\\ * &{} * &{} * &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$.

• (iv) Suppose the number of zero entries in U is four. If four columns and four rows have zero entries, then there is one ZED-equivalent class, i.e., $\begin{bmatrix} 0 &{} * &{} * &{} *\\ * &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} *\\ * &{} * &{} * &{} 0 \end{bmatrix}$.

  If four columns and three rows have zero entries, then there is one ZED-equivalent class, i.e., $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} *\\ * &{} * &{} * &{} 0\\ * &{} * &{} * &{} * \end{bmatrix}$.

  If four columns and two rows have two zero entries, then there is one ZED-equivalent class, i.e., $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} 0\\ * &{} * &{} * &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$.

  If three columns and three rows have zero entries, then there are two ZED-equivalent classes, i.e., $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ 0 &{} * &{} * &{} *\\ * &{} * &{} 0 &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$ and $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} *\\ * &{} * &{} 0 &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$.

  If three columns and two rows have zero entries, then there is one ZED-equivalent class, i.e., $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ 0 &{} * &{} 0 &{} *\\ * &{} * &{} * &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$.

  If two columns and two rows have zero entries, then there is one ZED-equivalent class, i.e., $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ 0 &{} 0 &{} * &{} *\\ * &{} * &{} * &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$.

  One can verify that only the first and the third ZED-equivalent classes of matrices can be unitary matrices, i.e., $\begin{bmatrix} 0 &{} * &{} * &{} *\\ * &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} *\\ * &{} * &{} * &{} 0 \end{bmatrix}$ and $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} 0\\ * &{} * &{} * &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$.

• (v) Suppose the number of zero entries in U is six. If four columns and four rows have zero entries, then there are eight ZED-equivalent classes, i.e., $\begin{bmatrix} * &{} 0 &{} 0 &{} 0\\ 0 &{} * &{} * &{} *\\ 0 &{} * &{} * &{} *\\ 0 &{} * &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ * &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} *\\ * &{} * &{} * &{} 0 \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ * &{} 0 &{} 0 &{} *\\ * &{} * &{} 0 &{} *\\ * &{} * &{} * &{} 0 \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ * &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} 0\\ * &{} * &{} * &{} 0 \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ 0 &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} *\\ * &{} * &{} * &{} 0 \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ * &{} 0 &{} * &{} *\\ 0 &{} * &{} 0 &{} *\\ * &{} * &{} * &{} 0 \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ * &{} 0 &{} * &{} *\\ * &{} 0 &{} 0 &{} *\\ * &{} * &{} * &{} 0 \end{bmatrix}$ and $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ * &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} *\\ * &{} * &{} 0 &{} 0 \end{bmatrix}$.

  If four columns and three rows have zero entries, then there are seven ZED-equivalent classes, i.e., $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} *\\ * &{} * &{} * &{} 0\\ * &{} * &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} * &{} * &{} 0\\ 0 &{} * &{} * &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} * &{} * &{} 0\\ * &{} 0 &{} * &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} * &{} * &{} 0\\ * &{} * &{} * &{} 0\\ * &{} * &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ 0 &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} 0\\ * &{} * &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ 0 &{} * &{} 0 &{} *\\ 0 &{} * &{} * &{} 0\\ * &{} * &{} * &{} * \end{bmatrix}$ and $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ 0 &{} * &{} 0 &{} *\\ * &{} 0 &{} * &{} 0\\ * &{} * &{} * &{} * \end{bmatrix}$.

  If four columns and two rows have zero entries, then there is one ZED-equivalent class, i.e., $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ * &{} 0 &{} 0 &{} 0\\ * &{} * &{} * &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$.

  If three columns and three rows have zero entries, then there are four ZED-equivalent classes, i.e., $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} *\\ 0 &{} * &{} * &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ 0 &{} * &{} 0 &{} *\\ 0 &{} 0 &{} * &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$ and $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ 0 &{} * &{} 0 &{} *\\ * &{} 0 &{} 0 &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$.

  If three columns and two rows have zero entries, then there is one ZED-equivalent class, i.e., $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} 0 &{} *\\ * &{} * &{} * &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$.

  One can verify that only one ZED-equivalent class of matrices can be unitary matrices, i.e., $\begin{bmatrix} * &{} 0 &{} 0 &{} 0\\ 0 &{} * &{} * &{} *\\ 0 &{} * &{} * &{} *\\ 0 &{} * &{} * &{} * \end{bmatrix}$.

• (vi) Suppose the number of zero entries in U is seven. If four columns and four rows have zero entries, then there are eight ZED-equivalent classes, i.e., $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} *\\ 0 &{} * &{} * &{} *\\ * &{} * &{} * &{} 0 \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} *\\ * &{} * &{} * &{} 0 \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} * &{} * &{} 0\\ 0 &{} * &{} * &{} *\\ * &{} 0 &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} * &{} * &{} 0\\ 0 &{} * &{} * &{} *\\ * &{} * &{} * &{} 0 \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} * &{} * &{} 0\\ * &{} 0 &{} * &{} *\\ * &{} 0 &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} * &{} * &{} 0\\ * &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} * &{} * &{} 0\\ * &{} 0 &{} * &{} *\\ * &{} * &{} * &{} 0 \end{bmatrix}$, and $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} * &{} * &{} 0\\ * &{} * &{} * &{} 0\\ * &{} * &{} * &{} 0 \end{bmatrix}$.

  If four columns and three rows have zero entries, then there are ten ZED-equivalent classes, i.e., $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} 0 &{} *\\ * &{} * &{} * &{} 0\\ * &{} * &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} 0\\ 0 &{} * &{} * &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} 0\\ * &{} * &{} 0 &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} 0\\ * &{} * &{} * &{} 0\\ * &{} * &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} *\\ 0 &{} * &{} * &{} 0\\ * &{} * &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} 0\\ * &{} * &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} * &{} * &{} 0\\ 0 &{} 0 &{} * &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} * &{} * &{} 0\\ 0 &{} * &{} * &{} 0\\ * &{} * &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} * &{} * &{} 0\\ * &{} 0 &{} 0 &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$ and $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} * &{} * &{} 0\\ * &{} 0 &{} * &{} 0\\ * &{} * &{} * &{} * \end{bmatrix}$.

  If three columns and three rows have zero entries, then there are three ZED-equivalent classes, i.e., $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} 0 &{} *\\ 0 &{} * &{} * &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} *\\ 0 &{} 0 &{} * &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$ and $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} *\\ 0 &{} * &{} 0 &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$.

  One can verify that there is only one ZED-equivalent class of matrices can be unitary matrices, i.e., $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} * &{} * &{} 0\\ * &{} * &{} * &{} 0\\ * &{} * &{} * &{} 0 \end{bmatrix}$.

  Other ZED-equivalent classes of matrices cannot be unitary matrices.

• (vii) Suppose the number of zero entries in U is eight. If four columns and four rows have zero entries, then there are sixteen ZED-equivalent classes, i.e., $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} 0 &{} *\\ 0 &{} * &{} * &{} *\\ * &{} * &{} * &{} 0 \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} 0 &{} *\\ * &{} * &{} * &{} 0\\ * &{} * &{} * &{} 0 \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} 0\\ 0 &{} * &{} * &{} *\\ * &{} 0 &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} 0\\ 0 &{} * &{} * &{} *\\ * &{} * &{} 0 &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} 0\\ * &{} * &{} 0 &{} *\\ * &{} * &{} 0 &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} 0\\ * &{} * &{} 0 &{} *\\ * &{} * &{} * &{} 0 \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} *\\ 0 &{} 0 &{} * &{} *\\ * &{} * &{} * &{} 0 \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} *\\ 0 &{} * &{} 0 &{} *\\ * &{} * &{} * &{} 0 \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} *\\ 0 &{} * &{} * &{} 0\\ * &{} 0 &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} *\\ 0 &{} * &{} * &{} 0\\ * &{} * &{} 0 &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} *\\ 0 &{} * &{} * &{} 0\\ * &{} * &{} * &{} 0 \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} * &{} * &{} 0\\ 0 &{} * &{} * &{} 0\\ * &{} 0 &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} * &{} * &{} 0\\ 0 &{} * &{} * &{} 0\\ * &{} * &{} * &{} 0 \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ 0 &{} 0 &{} * &{} *\\ 0 &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} 0 \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ 0 &{} 0 &{} * &{} *\\ 0 &{} * &{} 0 &{} *\\ * &{} 0 &{} * &{} 0 \end{bmatrix}$ and $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ 0 &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} 0\\ * &{} * &{} 0 &{} 0 \end{bmatrix}$.

  If four columns and three rows have zero entries, then there are three ZED-equivalent classes, i.e., $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} 0 &{} *\\ 0 &{} * &{} * &{} 0\\ * &{} * &{} * &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} 0\\ 0 &{} * &{} 0 &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$ and $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} 0\\ * &{} * &{} 0 &{} 0\\ * &{} * &{} * &{} * \end{bmatrix}$.

  If three columns and three rows have zero entries, then there is one ZED-equivalent class, i.e., $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} *\\ * &{} * &{} * &{} * \end{bmatrix}$. One can verify that there is only one ZED-equivalent class of matrices can be unitary matrices, i.e., $\begin{bmatrix} 0 &{} 0 &{} * &{} *\\ 0 &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} 0\\ * &{} * &{} 0 &{} 0 \end{bmatrix}$.

  Other ZED-equivalent classes of matrices cannot be unitary matrices.

• (viii) Suppose the number of zero entries in U is ten. If four columns and four rows have zero entries, then there are ten ZED-equivalent classes, i.e., $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} 0 &{} *\\ * &{} * &{} * &{} 0 \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} 0\\ * &{} * &{} 0 &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} 0\\ * &{} * &{} * &{} 0 \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} 0 \end{bmatrix}$. $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} 0 &{} *\\ 0 &{} * &{} * &{} 0\\ * &{} 0 &{} * &{} 0 \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} 0\\ 0 &{} 0 &{} * &{} *\\ * &{} * &{} 0 &{} 0 \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} 0\\ 0 &{} * &{} 0 &{} *\\ * &{} 0 &{} 0 &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} 0\\ 0 &{} * &{} 0 &{} *\\ * &{} 0 &{} * &{} 0 \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} 0\\ 0 &{} * &{} 0 &{} *\\ * &{} * &{} 0 &{} 0 \end{bmatrix}$ and $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} 0\\ * &{} * &{} 0 &{} 0\\ * &{} * &{} 0 &{} 0 \end{bmatrix}$.

  One can verify that there is only one ZED-equivalent class of matrices can be unitary matrices, i.e., $\begin{bmatrix} 0 &{} 0 &{} 0 &{} *\\ 0 &{} 0 &{} * &{} 0\\ * &{} * &{} 0 &{} 0\\ * &{} * &{} 0 &{} 0 \end{bmatrix}$.

  Other ZED-equivalent classes of matrices cannot be unitary matrices.

• (ix) Suppose the number of zero entries in U is twelve. Because U has full rank, there is only one ZED-equivalent class, i.e., $\begin{bmatrix} * &{} 0 &{} 0 &{} 0\\ 0 &{} * &{} 0 &{} 0\\ 0 &{} 0 &{} * &{} 0\\ 0 &{} 0 &{} 0 &{} * \end{bmatrix}$.

$\square$

Appendix G: The proof of Lemma 26

Proof

Suppose U is a unitary matrix in ${\mathbb {U}}^n$ and $n\ge 5$.

• (i) Suppose the number of zero entries in U is 1. One can verify that there is only one ZED-equivalent class, i.e., $\begin{bmatrix} 0 &{} * &{} \cdots &{} *\\ * &{} * &{} \cdots &{} *\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ * &{} * &{} \cdots &{} * \end{bmatrix}$.

• (ii) Suppose the number of zero entries in U is 2. One can verify that there are two ZED-equivalent classes, i.e., $\begin{bmatrix} 0 &{} 0 &{} \cdots &{} *\\ * &{} * &{} \cdots &{} *\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ * &{} * &{} \cdots &{} * \end{bmatrix}$ and $\begin{bmatrix} 0 &{} * &{} \cdots &{} *\\ * &{} 0 &{} \cdots &{} *\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ * &{} * &{} \cdots &{} * \end{bmatrix}$.

• (iii) Suppose the number of zero entries in U is 3. One can verify that there are four ZED-equivalent classes, i.e., $\begin{bmatrix} 0 &{} * &{} * &{} * &{} \cdots &{} *\\ * &{} 0 &{} * &{} * &{} \cdots &{} *\\ * &{} * &{} 0 &{} * &{} \cdots &{} *\\ * &{} * &{} * &{} * &{} \cdots &{} *\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ * &{} * &{} * &{} * &{} \cdots &{} * \end{bmatrix}$ $\begin{bmatrix} 0 &{} * &{} 0 &{} * &{} \cdots &{} *\\ * &{} 0 &{} * &{} * &{} \cdots &{} *\\ * &{} * &{} * &{} * &{} \cdots &{} *\\ * &{} * &{} * &{} * &{} \cdots &{} *\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ * &{} * &{} * &{} * &{} \cdots &{} * \end{bmatrix}$, $\begin{bmatrix} 0 &{} 0 &{} 0 &{} * &{} \cdots &{} *\\ * &{} * &{} * &{} * &{} \cdots &{} *\\ * &{} * &{} * &{} * &{} \cdots &{} *\\ * &{} * &{} * &{} * &{} \cdots &{} *\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ * &{} * &{} * &{} * &{} \cdots &{} * \end{bmatrix}$ and $\begin{bmatrix} 0 &{} 0 &{} * &{} * &{} \cdots &{} *\\ * &{} 0 &{} * &{} * &{} \cdots &{} *\\ * &{} * &{} * &{} * &{} \cdots &{} *\\ * &{} * &{} * &{} * &{} \cdots &{} *\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ * &{} * &{} * &{} * &{} \cdots &{} * \end{bmatrix}$.

• (iv) Suppose the number of zero entries in U is $n^2-n-5$. One can verify that there is only one ZED-equivalent class, i.e., $\begin{bmatrix} 0 &{} * &{} * &{} 0 &{} \cdots &{} 0\\ * &{} * &{} * &{} 0 &{} \cdots &{} 0\\ * &{} * &{} * &{} 0 &{} \cdots &{} 0\\ 0 &{} 0 &{} 0 &{} * &{} \cdots &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} * \end{bmatrix}$.

• (v) Suppose the number of zero entries in U is $n^2-n-4$. One can verify that there is only one ZED-equivalent class, i.e., $\begin{bmatrix} * &{} * &{} 0 &{} 0 &{} \cdots &{} 0\\ * &{} * &{} 0 &{} 0 &{} \cdots &{} 0\\ 0 &{} 0 &{} * &{} * &{} \cdots &{} 0\\ 0 &{} 0 &{} * &{} * &{} \cdots &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} \cdots &{} * \end{bmatrix}$.

• (vi) Suppose the number of zero entries in U is $n^2-n-2$. One can verify that there is only one ZED-equivalent class, i.e., $\begin{bmatrix} * &{} * &{} 0 &{} \cdots &{} 0\\ * &{} * &{} 0 &{} \cdots &{} 0\\ 0 &{} 0 &{} * &{} \cdots &{} 0\\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} * \end{bmatrix}$.

• (vii) Suppose the number of zero entries in U is $n^2-n$. One can verify that there is only one ZED-equivalent class, i.e., $\begin{bmatrix} * &{} 0 &{} \cdots &{} 0\\ 0 &{} * &{} \cdots &{} 0\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} \cdots &{} * \end{bmatrix}$.

$\square$