In order to have biological denitrification, one need anaerobic zones within the media (or substrate). There are types of aerobic bacteria, called facultative, which normally use free oxygen. But when there is none, they can switch to using nitrate They basically get the O out of the NO3 and that leaves nitrogen gas which is harmless and goes back into the air. To get this process to happen in a filter requires properly designed media. What happens is as the water passes through the media, the aerobic bacteria use up all the free oxygen. But they are then producing nitrate. The facultative aerobes, living deeper in the media, having no oxygen available, switch to using the nitrate. And then you have denitrification.
This same process typically occurs in the substrate of planted tanks with roots. In addition, plants themselves use nitrate as a nutrient. This is why high tech planted tanks often require the addition of nitrate because what the tank produces is not sufficient for the plant needs.
Encouraging natural biological denitrification in most filters requries specially designed media. In nature, the denitrifying process occurs because there is huge natural media volume. This is what is behind the great effectiveness of the Hamburg Matten filter which uses a huge amount of media. Here is an example I posted on another thread:
I have a 20 gal. long tank for growing out pleco youngsters. I am almost finished swapping out the 3 hang-ons that used to be on the tank. I had an AquaClear 150gph and a 100gph plus a Tetra Whisper 100 gph.
The Aquaclears each had two sponges and a layer of floss and the Whisper had a bit of floss and pieces of sponge in it.
2 sponges in the 150 @ 3.5 x 2.25 x 1.62 = 12.76 cu. in. x 2 = 25.52 cu. in.
2 sponges in the 100 @ 2.375 x 2.25 x 1.625 = 8.67 cu. in. x 2 = 17.36 cu. in
Pieces of sponge in the Whisper estimated to be = 10.00 cu. in
So the total volume of the sponge media in all three filters was about 53 cu. in.
These are all being replaced by a Matten filter that is 11.75 x 11.5 x 2.0 = 270.25 cu. in.
As you can see the Matten, w/o considering the PPI involved, is over 5 times the volume of the hang-ons combined. But if we were to calculate the available surface area for the bacteria etc., the amount of surface area in the Matten is going to be more like 10-20 times that of the other three filters combined. Even better, I am running that filter from a single air pump which draws less power than the three hang-ons did.
Now, as to the SeaChem Matrix and similar products. The theory is that it is constructed to have the proper porosity to encourage denitrification deeper within the media. This is not as effective as huger media volumes, but it should be more effective at encouraging denitrification than most media with its bigger pore sizes and fewer of them such as typical sponges or noodles. The key is getting the water to flow through media with much more bio-film inside to set up denitrification. Note, not all the water passes though the media where there are proper conditions for denitrifying. That means one is not removing all the oxygen from the water. But we also have the surface agitation in a tank to replace that oxygen as fast as it may be used. Incidentally, biological filtration works best at slower flow rates than most filters have. It is better to use other equipment to create circulation. That slower flow helps with all the biological filtration processes. Faster flow rates mean less comprehensive filtration which we can easily overcome with our regular water changes, media rinsing, floss replacement and substrate vacuuming.
SeaChem is not the only company to make media specifically designed to host denitrification in filters. Sera would be another with their Siporax. These products can be added to almost any filter to encourage denitrification. However, in larger scale application I believe there are more cost effective methods. The average filters we use on many tanks are usually too small to allow for sufficient media to do all the work that is potentially possible when using a greater volume of media. Even using those products there may not be sufficient media to host all of the potential bacteria required for optimal filtration. But, if one can lowerenough of one's nitrate this way, that may be enough to solve many problems.
As for Purigen, it does similar (but not identical) things as carbon, but it is rechargeable. It is also more expensive. It is quite good at removing organics. It was suggested I use it in my altum angel tank as it is felt that these fish communicate chemically/homonally at times and that the Purigen would reduce the potential build up of these things in a tank. I have never used it because my altum tank is tea stained water and both carbon and Purigen would remove the staining.
One last comment here. All of the above involves biological/natural means for achieving our goals for keeping our tanks clean, safe and healthy for our fish and critters. There is also an assortment of chemical means that are supposed to do a lot of the same things. However, it is almost always preferable to do things the way nature "designed' them than to seek shortcuts using chemicals. Most of these come with either side effects or they create of something else that maysimply be not as bad as what it "solved." Examples would be having nitrifying bacteria to handle ammonia instead of constantly using an ammonia detoxifying product, or using ro water to soften water and/or tp lower pH instead of buffers or ion exchange softeners which add sodium to the water etc.