monomial matrices

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I have matrices of the form
$ A= (F_N otimes I_M)Pi Sigma (F_N^dagger otimes I_M)$, where $Sigma$ is diagonal whose elements are the powers of $exp(2pi j /(MN))$ from exponent $0$ to exponent $MN-1$, and hence having unit magnitude, $Pi$ is a permutation matrix, $I$ is the identity matrix, and $F_N$ is the $N$-point discrete-Fourier-transform matrix. $A$ is unitary. Direct numerical calculations show that $A$ is a monomial matrix. How can I prove this?




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    I have matrices of the form
    $ A= (F_N otimes I_M)Pi Sigma (F_N^dagger otimes I_M)$, where $Sigma$ is diagonal whose elements are the powers of $exp(2pi j /(MN))$ from exponent $0$ to exponent $MN-1$, and hence having unit magnitude, $Pi$ is a permutation matrix, $I$ is the identity matrix, and $F_N$ is the $N$-point discrete-Fourier-transform matrix. $A$ is unitary. Direct numerical calculations show that $A$ is a monomial matrix. How can I prove this?




    matrices




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      up vote
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      down vote

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      I have matrices of the form
      $ A= (F_N otimes I_M)Pi Sigma (F_N^dagger otimes I_M)$, where $Sigma$ is diagonal whose elements are the powers of $exp(2pi j /(MN))$ from exponent $0$ to exponent $MN-1$, and hence having unit magnitude, $Pi$ is a permutation matrix, $I$ is the identity matrix, and $F_N$ is the $N$-point discrete-Fourier-transform matrix. $A$ is unitary. Direct numerical calculations show that $A$ is a monomial matrix. How can I prove this?




      matrices




      share|cite|improve this question













      I have matrices of the form
      $ A= (F_N otimes I_M)Pi Sigma (F_N^dagger otimes I_M)$, where $Sigma$ is diagonal whose elements are the powers of $exp(2pi j /(MN))$ from exponent $0$ to exponent $MN-1$, and hence having unit magnitude, $Pi$ is a permutation matrix, $I$ is the identity matrix, and $F_N$ is the $N$-point discrete-Fourier-transform matrix. $A$ is unitary. Direct numerical calculations show that $A$ is a monomial matrix. How can I prove this?





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