$X≔\mathrm{Array}\left(\left[1\,2\mathrm{I}\,3\right]\right)$

$X≔\left[\begin{array}{ccc}1& 2\mathrm{I}& 3\end{array}\right]$

$\mathrm{RootMeanSquare}\left(X\right)$

$2.16024689946928694$

$Y≔\left[3\mathrm{sqrt}\left(2\right)\,4\mathrm{sqrt}\left(2\right)\right]$

$Y≔\left[3\sqrt{2}\,4\sqrt{2}\right]$

$\mathrm{RootMeanSquare}\left(Y\right)$

$5.$

$Z≔\mathrm{Matrix}\left(\left[\left[5\,-10\right]\,\left[15\,20+\mathrm{I}\right]\right]\right)$

$Z≔\left[\begin{array}{cc}5& \mathrm{-10}\\ 15& 20+\mathrm{I}\end{array}\right]$

$\mathrm{RootMeanSquare}\left(Z\right)$

$13.7021896060447208$

Parseval's Theorem shows that the root mean square of the Discrete Fourier Transform (DFT) of a signal is the same as that of the original signal. For example:

$A≔\mathrm{LinearAlgebra}:-\mathrm{RandomVector}\left(5\,\mathrm{datatype}=\mathrm{complex}\left[8\right]\right)$

$A≔\left[\begin{array}{c}\mathrm{-94.}-58.\mathrm{I}\\ 12.-7.\mathrm{I}\\ 21.-53.\mathrm{I}\\ 40.-25.\mathrm{I}\\ 43.+97.\mathrm{I}\end{array}\right]$

$B≔\mathrm{Vector}\left[\mathrm{column}\right]\left(\mathrm{DFT}\left(A\right)\right)$

$B≔\left[\begin{array}{c}9.83869910099907-20.5718253929981\mathrm{I}\\ \mathrm{-108.101349850469}+32.8995037690910\mathrm{I}\\ \mathrm{-68.9361787597612}-69.2123797669952\mathrm{I}\\ \mathrm{-38.0783657332284}-69.3476788839996\mathrm{I}\\ \mathrm{-4.91319464252087}-3.45956242008582\mathrm{I}\end{array}\right]$

$\mathrm{rms\_\_A}≔\mathrm{RootMeanSquare}\left(A\right)$

$\mathrm{rms\_\_A}≔76.3229978446863555$

$\mathrm{rms\_\_B}≔\mathrm{RootMeanSquare}\left(B\right)$

$\mathrm{rms\_\_B}≔76.3229978446863555$

We can also compare the original signal with the Inverse Discrete Fourier Transform (IDFT) of its DFT:

$C≔\mathrm{Vector}\left[\mathrm{column}\right]\left(\mathrm{InverseDFT}\left(B\right)\right)$

$C≔\left[\begin{array}{c}\mathrm{-94.0000000000000}-58.\mathrm{I}\\ 12.0000000000000-7.00000000000001\mathrm{I}\\ 21.0000000000000-53.\mathrm{I}\\ 40.0000000000000-25.0000000000000\mathrm{I}\\ 43.+97.0000000000000\mathrm{I}\end{array}\right]$

$\mathrm{RootMeanSquare}\left(A-C\right)$

$1.93451702870987411\times {10}^{\mathrm{-14}}$