Assume that the numbers of transmitting antennas, receiving antennas, and delayed waves are $N_t$, $N_r$, and $N_p$, respectively. The channel matrix is three-dimensional ($N_r$, $N_t$, $N_p$).
In eigenmode transmission, when $N_p=1$, the channel matrix is subjected to singular value decomposition.
How to perform eigenmode transmission when $N_p>1$?
Assuming $N_p$ is, as discussed in the comments, the number of taps in a multipath channel.
While higher-order singular value decomposition on tensors with more than two dimensions are a thing, the fact that the eigenvectors turn out to be tensors themselves makes it not as desirable as method of MIMO precoding.
Instead, the problem is solved in two steps: The "multipath" dimension describes the channel impulse response between a transmit and a receive antenna. You thus start by looking for common eigenvectors of all the convolution matrices that arises when you convolve with these channel impulse responses.
Luckily, convolution is an LTI system, so all these channels have the same set of eigenvectors, namely, harmonic oscillations.
So, you first decompose the signals you might want to send into harmonic oscillations: Exactly what the (discrete, since we're discrete in time here) Fourier transform does! So, you use the IDFT as a means to transform each possible transmit signal onto an ONB of eigenvectors of (sampled) harmonic oscillations.
Ok, so we know that when we apply the "$N_p$-direction" aspect of out channel tensor to our thus decomposed signal, we get the input out, changed by nothing but a multiplication with a scalar (the individual Fourier coefficient of the channel impulse response). The IDFT transforms each of our $N_r\times N_t$ multipath channels (each of $N_p$ components) into a set of completely independent scalar channels! Thus, you reduce the problem of an $N_r\times N_t \times N_p$ channel into a problem of $N_p$ independent $N_r\times N_t (\times 1)$ channels. Much, much easier, and lossless! We already have a solution for these channels, so we're done.
Technically, this decomposition is most commonly done via OFDM – in fact, I've never seen a practically employed wideband (i.e., $N_p > 1$) MIMO system that wasn't based on relatively classical OFDM. So, you divide your transmit signal(s) into individual frequency components using OFDM narrowband enough that each of them only sees a single-tap (i.e., frequency-flat) channel, then for these narrow band components, each "sees" a different $N_r\times N_t$ matrix, and you do "normal" MIMO on each of them.
OFDM is probably a technology you would want to read into if you're working with MIMO.
