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Generalized Tenney norms and Tp interval space: Difference between revisions

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<math>\displaystyle
<math>\displaystyle
\left[\begin{array}{rrrrrl}
\begin{bmatrix}
\tmonzo{1 & 0 & 0 & 0} \\
1 & 0 & 0 & 0 \\
\tmonzo{0 & 2 & 0 & -1} \\
0 & 2 & 0 & -1 \\
\tmonzo{0 & -1 & 1 & 0} \\
0 & -1 & 1 & 0 \\
\end{array}\right]</math>
\end{bmatrix}</math>


Note that the "rows" here are written in kets; this is a convention to signify that each ket, representing a monzo, is actually supposed to represent a column of the matrix as explained in [[Subgroup basis matrices]].
Note that the "rows" here are written in kets; this is a convention to signify that each ket, representing a monzo, is actually supposed to represent a column of the matrix as explained in [[Subgroup basis matrices]].
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<math>\displaystyle  
<math>\displaystyle  
\lVert \vec m \rVert_{\text{T} 1}^{2 \text{.} 9/7 \text{.} 5/3} = \left \lVert
\left\lVert \vec m \right\rVert_{\text{T} 1}^{2 \text{.} 9/7 \text{.} 5/3} = \left \lVert
\begin{bmatrix}
\begin{bmatrix}
\log_2(2) & 0 & 0 & 0\\
\log_2(2) & 0 & 0 & 0\\
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0 & 0 & \log_2(5) & 0\\
0 & 0 & \log_2(5) & 0\\
0 & 0 & 0 & \log_2(7)
0 & 0 & 0 & \log_2(7)
\end{bmatrix} \cdot \left[ \begin{array}{rrrrrl}
\end{bmatrix} \cdot \begin{bmatrix}
\tmonzo{1 & 0 & 0 & 0} \\
1 & 0 & 0 & 0 \\
\tmonzo{0 & 2 & 0 & -1} \\
0 & 2 & 0 & -1 \\
\tmonzo{0 & -1 & 1 & 0} \\
0 & -1 & 1 & 0 \\
\end{array} \right] \cdot \left[ \begin{array}{rrrrl}
\end{bmatrix} \cdot \begin{bmatrix}
\tmonzo{0 & -2 & 1} \\
0 & -2 & 1 \\
\end{array} \right]
\end{bmatrix}
\right \rVert_1
\right \rVert_1
</math>
</math>
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which finally resolves to
which finally resolves to


<math>\displaystyle \lVert \vec m \rVert_{\text{T} 1}^{2.9/7.5/3} = \lVert \monzo{ \begin{matrix} 0 & -7.925 & 2.322 & 5.615 \end{matrix} } \rVert_1 = 15.861</math>
<math>\displaystyle \left\lVert \vec m \right\rVert_{\text{T} 1}^{2.9/7.5/3} = \lVert \monzo{ \begin{matrix} 0 & -7.925 & 2.322 & 5.615 \end{matrix} } \rVert_1 = 15.861</math>


Note that {{nowrap|15.861 {{=}} {{!}}0{{!}} + {{!}}−7.925{{!}} + {{!}}2.322{{!}} + {{!}}5.615{{!}}}}, which is the ''L''<sub>1</sub> norm of the vector.
Note that {{nowrap|15.861 {{=}} {{!}}0{{!}} + {{!}}−7.925{{!}} + {{!}}2.322{{!}} + {{!}}5.615{{!}}}}, which is the ''L''<sub>1</sub> norm of the vector.
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