# Matrix factorisation of the Fourier matrix

I am currently reading a paper Low Communication FMM-Accelerated FFT on GPUs

In that I am not able to understand the definition of the twiddle factor matrix $$T_{P, M}$$.

The Fourier matrix $$F_N$$ is defined as:

$$[F_N]_{i, j} = \omega_N^{ij}$$

According to the paper, the Fourier matrix is factorised as :

$$F_N = \Pi_{M, P}(I_M \otimes F_P) \Pi_{P, M} T_{P, M} (I_P \otimes F_M) \Pi_{M, P}$$

Where $$\Pi$$ is the block to cyclic operator.

The paper defines $$T_{P,M}$$ as the diagonal matrix of the twiddle factors.

Mathematically:

$$[T_{P, M}]_{i, j} = \delta_{i, j} \omega_{N}^{(i\mod M) \cdot \lfloor i/M \rfloor}$$

I feel that the above equation is wrong. Because if the matrix is diagonal, then $$i < min(P, M)$$ meaning that $$\lfloor i/M \rfloor$$ will always be 0. This would mean that the $$T$$ is a diagonal matrix of 1s only.

Can someone help me figure out the correct equation for $$T$$?