Hmm, perhaps you don't understand what "lossless matrixing" means. It means changing the co-ordinate system to express a multi-dimensional quantity in a more convenient way. Any n
-dimensional vector can be represented using any n
axes, as long as they're independent.
So, for example, 2-channel stereo can be stored either as
1) "Left" and "Right"; or
2) "Left+Right" and "Left-Right"
The two are equivalent, and can be losslessly converted between, but the second form has the advantage that mono playback can be achieved with only the first channel. Also, the second channel will often be simpler to encode - MP3 etc often use this form ("joint stereo"). As does (did?) lots of general sound production in TV etc, I understand.
Consider also television. You could use:
1) Red, Green, Blue; or
2) Luma(R+G+B), Red Chroma(R-Luma), Blue Chroma(B-Luma)
All modern video uses the latter; monochrome playback only requires 1 channel, and luma and chroma can be compressed very differently because of the way the eye responds to them. It then gets losslessly transformed back to RGB for the display.
This MLP downmix system is exactly analogous. 6 channels of information can be stored as:
1) L, R, C, LFE, LS, RS; or
LFE, C, LFE, LS, RS
The latter 6-channel form has the handy advantage that the first two channels are a ready-made stereo downmix, with parameters a
defined by the mixer. But just like the other cases, you can transform back to the original co-ordinate system.
In theory, the maths to reconstruct the original left and right is as simple as:
It's only rounding errors you have to worry about.
Actually, I've just realised that as the right channel can be included in the left downmix, and vice versa, the maths is more complicated - you've got to solve a pair of simultaneous equations. Still, simple enough though, but I'm not going to attempt to write it down here.