Let $F=(w^{kl})_{k,l=0}^{n-1}$ be the discrete Fourier matrix of size $n$ where $w=\exp^{-\frac{2\pi i}{n}}$.

It is a well-known that $F_n^4 = I_n$ where $I_n$ represents the identity matrix of dimension $n$. This leads to the conclusion that $F$ possess precisely four eigenvalues, namely $\{\pm1,\pm i\}$, when $n\geq4$. Let $E_{\lambda}$ be the linear space consisting of all eigenvectors of $F$ corresponding to the eigenvalue $\lambda$.

Let $P_{\lambda}$ be the projection onto $E_{\lambda}$. It is known that:

$P_1=\frac{1}{4}(F^3+F^2+F+I)$

$P_{-1}=\frac{1}{4}(-F^3+F^2-F+I)$

$P_{i}=\frac{1}{4}(iF^3-F^2-iF+I)$

$P_{-i}=\frac{1}{4}(-iF^3-F^2+iF+I)$

Obviously $P_{\lambda}$s are mutually orthogonal and $\sum P_{\lambda}=I$. Therefore any signal $x$ is decomposed into four orthgonal components

$$x=P_1x+P_{-1}x+P_{i}x+P_{-i}x$$

If $y =P_{\lambda}x$, then $Fy=\lambda y$ is a direct consequence.

Q. What potential applications and advantages can be derived from this decomposition?

Let $F=(w^{kl})_{k,l=0}^{n-1}$ be the discrete Fourier matrix of size $n$ where $w=\exp\left(-\frac{2\pi i}{n}\right)$.

It is a well-known that $F_n^4 = I_n$ where $I_n$ represents the identity matrix of dimension $n$. This leads to the conclusion that $F$ possess precisely four eigenvalues, namely $\{\pm1,\pm i\}$, when $n\geq4$. Let $E_{\lambda}$ be the linear space consisting of all eigenvectors of $F$ corresponding to the eigenvalue $\lambda$.

Let $P_{\lambda}$ be the projection onto $E_{\lambda}$. It is known that:

$P_1=\frac{1}{4}(F^3+F^2+F+I)$

$P_{-1}=\frac{1}{4}(-F^3+F^2-F+I)$

$P_{i}=\frac{1}{4}(iF^3-F^2-iF+I)$

$P_{-i}=\frac{1}{4}(-iF^3-F^2+iF+I)$

Obviously $P_{\lambda}$s are mutually orthogonal and $\sum P_{\lambda}=I$. Therefore any signal $x$ is decomposed into four orthgonal components

$$x=P_1x+P_{-1}x+P_{i}x+P_{-i}x$$

If $y =P_{\lambda}x$, then $Fy=\lambda y$ is a direct consequence.

Q. What potential applications and advantages can be derived from this decomposition?