Projection: Difference between revisions

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The answer comes in the form of a tuple. The first element is a list of our scaling factors (eigenvalues), and the second element is a list of our unrotated vectors (eigenvectors). So, this result is telling us that for scaling factors 1, 1, and 0, respectively, we have corresponding unrotated vectors {{vector|0 0 1}}, {{vector|1 0 0}}, and {{vector|4 -4 1}}. The set of eigenvectors with eigenvalue 0 constitute a comma basis, while the set of eigenvectors with eigenvalue 1 constitute an unchanged-interval basis (so, quarter-comma meantone is characterized by an unchanged <math>\frac21</math> and <math>\frac51</math>).

The list ~~of unrotated vectors~~ may be treated as a matrix <math>\mathrm{V}</math>, and the list of scaling factors may be diagonalized (placed along the main diagonal of an otherwise all-zeros matrix) as a matrix <math>\textit{Λ}</math> (that's a capital lambda, the same Greek letter we use the lowercase version of for the individual scaling factors).

The '''unrotated vector list''' may be treated as a matrix <math>\mathrm{V}</math>, and the list of scaling factors may be diagonalized (placed along the main diagonal of an otherwise all-zeros matrix) as the '''scaling factor matrix''' <math>\textit{Λ}</math> (that's a capital lambda, the same Greek letter we use the lowercase version of for the individual scaling factors).

As a bonus, we can get back to the projection via <math>P = \mathrm{V}\textit{Λ}\mathrm{V}^{-1}</math>.

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 The answer comes in the form of a tuple. The first element is a list of our scaling factors (eigenvalues), and the second element is a list of our unrotated vectors (eigenvectors). So, this result is telling us that for scaling factors 1, 1, and 0, respectively, we have corresponding unrotated vectors {{vector|0 0 1}}, {{vector|1 0 0}}, and {{vector|4 -4 1}}. The set of eigenvectors with eigenvalue 0 constitute a comma basis, while the set of eigenvectors with eigenvalue 1 constitute an unchanged-interval basis (so, quarter-comma meantone is characterized by an unchanged <math>\frac21</math> and <math>\frac51</math>).
-The list of unrotated vectors may be treated as a matrix <math>\mathrm{V}</math>, and the list of scaling factors may be diagonalized (placed along the main diagonal of an otherwise all-zeros matrix) as a matrix <math>\textit{Λ}</math> (that's a capital lambda, the same Greek letter we use the lowercase version of for the individual scaling factors).
+The '''unrotated vector list''' may be treated as a matrix <math>\mathrm{V}</math>, and the list of scaling factors may be diagonalized (placed along the main diagonal of an otherwise all-zeros matrix) as the '''scaling factor matrix''' <math>\textit{Λ}</math> (that's a capital lambda, the same Greek letter we use the lowercase version of for the individual scaling factors).
 As a bonus, we can get back to the projection via <math>P = \mathrm{V}\textit{Λ}\mathrm{V}^{-1}</math>.