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Cmloegcmluin (talk | contribs) use descriptive names for the two methods |
Cmloegcmluin (talk | contribs) →Method using the Smith Normal Form: clarify, because diagonal matrices are typically square |
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where <math>D</math> is the Smith Normal Form of <math>M</math>, which is a diagonal matrix ( | where <math>D</math> is the Smith Normal Form of <math>M</math>, which is a rectangular diagonal matrix (the name 'D' is for "diagonal"). Furthermore, if <math>M</math> was defactored, then not only is <math>D</math> diagonal (only has nonzero values along its main diagonal) but the numbers along its main diagonal are all equal to 1. This type of matrix is called an orthonormal matrix, and has the property that <math>D^{T}D = I</math>, which we're going to take advantage of. | ||
So first let's solve the decomposition equation for <math>M</math>. First, left-multiply by <math>L^{-1}</math>: | So first let's solve the decomposition equation for <math>M</math>. First, left-multiply by <math>L^{-1}</math>: | ||