Convolution matrix for converting 1080i to DVD? It's pretty common to shoot in HD but deliver on DVD. This requires a reduction in resolution from 1920x1080 to 720x480 (for NTSC).  If you just interpolate and throw away pixels, you get bad aliasing artifacts such as shimmering lines.  The right way to do it is to first apply a low-pass filter, then get rid of the extra pixels.

Most of the popular video editing suites (Final Cut Pro, Premiere Pro, etc.) will convert the resolution but they don't do the low-pass filter step.  You have to do it yourself.  Many people approximate it with a Gaussian blur filter, but that is pretty crude, hard to control and can easily lead to loss of detail.

A better approach would be to apply a 3x3 convolution filter, with the parameters precisely tuned to the specifics of this conversion.  Someone skilled in DSP theory should be able to derive the optimal convolution matrix once and for all so we can all use it.  I'm hoping someone can do so in answer to this question.
Really nice question, common problem.
After Effects, Autodesk Combustion and Autodesk Smoke can bring you a software downconversion(1920x1080 -> 720x480) with more quality than Final Cut Pro or Premiere Pro. Smoke can make it near perfectly. 

You can use (buy/rent) hardware downconverters, like AJA Kona 3, or Blackmagic clones. Better results goes with AJA. 

There are even more expensive/professionals solutions like Sonic and others which will make the MPEG-2 compression directly in hardware too.

If  you don't get your matrix filter from someone (not me, I'm not skilled for this kind of mission), best approach will be small tests with software and settings, tests, tests, tests, and then setup the workflow to not change forever!

Good Luck!