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?