Communications of the ACM
nraveling the Matrix
In a banded matrix, all the nonzero entries cluster around the diagonal.
Credit: Christine Daniloff / MIT News
Massachusetts Institute of Technology professor Gilbert Strang has developed a way to split certain types of matrices into simpler ones, which could help produce better video and audio data processing software or create smaller digital files.
Strang's method applies to banded matrices, in which almost all of the numbers are zeroes, except for those along diagonal bands at or near the center of the matrix. Since about 99 percent of the entries in a banded matrix are zeroes, multiplying it by another matrix is a very efficient process. After a signal has been processed, it must be converted back to its original form, which requires multiplying it by the inverse of the processing matrix. Unfortunately, the inverse of a banded matrix is almost always "full," meaning that almost all of the entries are nonzero, in which case all the speed advantages offered by banded matrices would be lost if restoring the signal required multiplying it by a full matrix.
Strang's technique for breaking up banded matrices into simpler ones makes it easier to tell whether these simpler matrices have banded inverses. The method enables engineers to determine whether a promising new signal-processing matrix will be practical.
From MIT News
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