Extended bra–ket notation: Difference between revisions
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They also considered using curved angle brackets <span style="font-family: STIXGeneral, 'Cambria Math', 'STIX Two Math', serif">⧼</span>…] […<span style="font-family: STIXGeneral, 'Cambria Math', 'STIX Two Math', serif">⧽</span>, (Unicode U+29FC and U+29FD), however in some fonts these are almost indistinguishable from ordinary angle brackets. If you see distinctly-curved angle brackets in the preceding sentence, it may only be because they have been specified to be shown in one of the fonts 'STIXGeneral', 'Cambria Math' (Windows) or 'STIX Two Math' (Mac). Here's an ordinary angle bracket followed by a curved angle bracket without special treatment: ⟩⧽. If this situation should change in future, a mnemonic for the curved angle brackets is that they can also be called "rounded" angle brackets, and a rounded number is an approximate number in the same way that a tempered interval is an approximation of a just interval. Curly brackets would still be used as a substitute in ASCII-only environments. | They also considered using curved angle brackets <span style="font-family: STIXGeneral, 'Cambria Math', 'STIX Two Math', serif">⧼</span>…] […<span style="font-family: STIXGeneral, 'Cambria Math', 'STIX Two Math', serif">⧽</span>, (Unicode U+29FC and U+29FD), however in some fonts these are almost indistinguishable from ordinary angle brackets. If you see distinctly-curved angle brackets in the preceding sentence, it may only be because they have been specified to be shown in one of the fonts 'STIXGeneral', 'Cambria Math' (Windows) or 'STIX Two Math' (Mac). Here's an ordinary angle bracket followed by a curved angle bracket without special treatment: ⟩⧽. If this situation should change in future, a mnemonic for the curved angle brackets is that they can also be called "rounded" angle brackets, and a rounded number is an approximate number in the same way that a tempered interval is an approximation of a just interval. Curly brackets would still be used as a substitute in ASCII-only environments. | ||
== Equivalent in matrix notation == | |||
In ordinary matrix notation, a matrix can be written in terms of its rows and columns using horizontal or vertical bars.<ref>L.N. Trefethen and D. Bau, III, [https://www.stat.uchicago.edu/~lekheng/courses/309/books/Trefethen-Bau.pdf Numerical linear algebra]. Society for industrial and applied mathematics, 1997.</ref> | |||
For example, the meantone mapping above can be written in terms of its rows as | |||
<math> | |||
\begin{bmatrix} | |||
1 & 0 & -4 \\ | |||
0 & 1 & 4 \\ | |||
\end{bmatrix} | |||
= | |||
\begin{bmatrix} | |||
1 & 0 & -4 \\ | |||
\hline | |||
0 & 1 & 4 \\ | |||
\end{bmatrix} | |||
</math> | |||
or symbolically, following the notation above, as | |||
<math> | |||
\begin{bmatrix} | |||
\hspace{1.45em} 𝒎_1 \hspace{1.45em} \\ | |||
\hline | |||
𝒎_2 | |||
\end{bmatrix} | |||
</math> | |||
Likewise, the comma basis above can be written in terms of its columns as | |||
<math> | |||
\left[ | |||
\begin{array}{cc} | |||
-11 & -7 \\ | |||
7 & 3 \\ | |||
0 & 1 | |||
\end{array} | |||
\right] | |||
= | |||
\left[ | |||
\begin{array}{c|c} | |||
-11 & -7 \\ | |||
7 & 3 \\ | |||
0 & 1 | |||
\end{array} | |||
\right] | |||
</math> | |||
or symbolically as | |||
<math> | |||
\left[ | |||
\begin{array}{c|c} | |||
& \\ | |||
\textbf{c}_1 & \textbf{c}_2 \\ | |||
& \\ | |||
\end{array} | |||
\right] | |||
</math> | |||
This notation is widely understood and can also express matrices with block structure by combining horizontal and vertical bars. | |||
== Notes == | == Notes == | ||