Fourier transform of a matrix!

I have a non-trivial task of computing the Fourier transform of a matrix $$\text{mat}[\alpha,\beta, t]$$ such that the Fourier transform looks like $$\text{matFT}[\alpha,\beta, \omega]$$. With $$\text{mat}[\alpha,\beta, t]$$ given by

 mat[α_, β_, t_] = {
{(Sqrt[α^2 - 4 β^2] Cosh[1/4 t Sqrt[α^2 - 4 β^2]] - α Sinh[1/4 t Sqrt[α^2 - 4 β^2]])^2/(-4 β^2 + α^2 Cosh[1/2 t Sqrt[α^2 - 4 β^2]] - α Sqrt[α^2 - 4 β^2] Sinh[1/2 t Sqrt[α^2 - 4 β^2]]),
(I β (-α + α Cosh[1/2 t Sqrt[α^2 - 4 β^2]] - Sqrt[α^2 - 4 β^2] Sinh[1/2 t Sqrt[α^2 - 4 β^2]]))/(4 β^2 - α^2 Cosh[1/2 t Sqrt[α^2 - 4 β^2]] + α Sqrt[α^2 - 4 β^2] Sinh[1/2 t Sqrt[α^2 - 4 β^2]])
},

{-((I β (-α + α Cosh[1/2 t Sqrt[α^2 - 4 β^2]] - Sqrt[α^2 - 4 β^2] Sinh[1/2 t Sqrt[α^2 - 4 β^2]]))/(4 β^2 - α^2 Cosh[1/2 t Sqrt[α^2 - 4 β^2]] + α Sqrt[α^2 - 4 β^2] Sinh[1/2 t Sqrt[α^2 - 4 β^2]])),
-((4 β^2 Sinh[1/4 t Sqrt[α^2 - 4 β^2]]^2)/(4 β^2 - α^2 Cosh[1/2 t Sqrt[α^2 - 4 β^2]] + α Sqrt[α^2 - 4 β^2] Sinh[1/2 t Sqrt[α^2 - 4 β^2]]))
}
};


How can this be achieved in Mathematica?

• Are you familiar with the commands FourierMatrix and/or FourierMatrix?
– user49048
Commented May 13, 2021 at 9:28
• @DiSp0sablE_H3r0 isn't FourierMatrix just the 1D Fourier transform though - a matrix of the Fourier basis which you dot with a 1d vector? It's not clear if OP wants multiple 1D or a single 2D Fourier transform. Commented May 13, 2021 at 10:50
• @flinty as you pointed out in the last sentence, it's not clear and this is why I wrote it just as a mere suggestion.
– user49048
Commented May 13, 2021 at 10:51