I think I explained this on another forum. but i'll put it here.

Very briefly, and I am by no means an EE, so someone like Bruno will probably have a much better explaination/definition... But the way I learned 10 years ago when I was in school was that...

All filters impart phase shift. The name implies how they phase shift.

A linear phase shift is just that, the phase shift from 0 to 180 degrees is linear across the frequency spectrum.

A non-linear phase shift could be anything from an exponential curve across the spectrum or it could be wobby with a much more complex pattern.

A minimum phase shift is a filter that tries not to shift the phase at all... and so the curve will either look like a flat line with a few bumps/ripples in it or possibly flat across with a drop straight down to 180 degree shift (makes a right angle on the graph, no slope) in the middle, or a combination of both.

So, a digital linear-phase filter would not be useful for correcting room modes since you will be shifting the phase around the center frequency linearly, and would be pretty audible.

A minimum-phase filter would try to leave as much of the original signal intact and have the least amount of phase shift around the center frequency possible, which would (hopefully) be less audible.

Again, I'm not an EE, nor do i claim to be any type of electric savvy person... i did take some classes in college a while ago and this is how I remember it.. which is to say, i could still be TOTALLY wrong... so please take this for what it's worth ($0.02).