Extended bra–ket notation: Difference between revisions

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One catch is that some matrices in RTT are more naturally thought of in terms of their ''rows'', while other matrices are more naturally thought of in terms of their ''columns''. EBK offers a good solution to this problem, though: it allows us to notate matrices ''either'' in terms of their rows ''or'' in terms their columns.

EBK is capable of doing this because of how vectors and covectors are interpreted as matrices. Specifically, a vector can be thought of as a <math>k</math>×1 matrix (many rows, one column), while a covector can be thought of as a 1×<math>k</math> matrix (one row, many columns).

EBK is capable of doing this because of how vectors and covectors are interpreted as matrices. Specifically, a vector can be thought of as a <math>(x,1)</math>-shaped matrix (many rows, one column), while a covector can be thought of as a <math>(1,x)</math>-shaped matrix (one row, many columns).

So, one way to think of a matrix would be as a single vector column which contains a number of covector rows. Another way to think of a matrix would be as a single covector row which contains a number of vector columns. Either way is possible, but in most cases, one of these two ways will be more easier to understand.

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RTT mappings are typically thought of in terms of their ''rows''. This mapping <math>M</math> has two rows; following mathematical conventions, let's call them <math>𝒎_1</math> and <math>𝒎_2</math>. And so to notate this mapping in EBK, we can first imagine capturing the rows as bras like we would normally: <math>𝒎_1</math> = {{map|1 0 -4}} and <math>𝒎_2</math> = {{map|0 1 4}}. Then, to put them together, we can think of this matrix as a single column containing these two rows, or in other words, a ket containing the two bras: {{ket|<math>𝒎_1</math> <math>𝒎_2</math>}}, or fully written out, {{ket|{{map|1 0 -4}} {{map|0 1 4}}}}.

For another example, the canonical [[comma basis]] for 7-ET consists of the two commas 2187/2048 and 135/128, with PC-vectors {{vector|-11 7}} and {{vector|-7 3 1}}, respectively. As a matrix <math>C</math>, we'd see this as:

For another example, the canonical [[comma basis]] for 7-ET consists of the two commas 2187/2048 and 135/128, with PC-vectors {{vector|-11 7}} and {{vector|-7 3 1}}, respectively. As a matrix <math>\mathrm{C}</math>, we'd see this as:

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 One catch is that some matrices in RTT are more naturally thought of in terms of their ''rows'', while other matrices are more naturally thought of in terms of their ''columns''. EBK offers a good solution to this problem, though: it allows us to notate matrices ''either'' in terms of their rows ''or'' in terms their columns.
-EBK is capable of doing this because of how vectors and covectors are interpreted as matrices. Specifically, a vector can be thought of as a <math>k</math>×1 matrix (many rows, one column), while a covector can be thought of as a 1×<math>k</math> matrix (one row, many columns).
+EBK is capable of doing this because of how vectors and covectors are interpreted as matrices. Specifically, a vector can be thought of as a <math>(x,1)</math>-shaped matrix (many rows, one column), while a covector can be thought of as a <math>(1,x)</math>-shaped matrix (one row, many columns).
 So, one way to think of a matrix would be as a single vector column which contains a number of covector rows. Another way to think of a matrix would be as a single covector row which contains a number of vector columns. Either way is possible, but in most cases, one of these two ways will be more easier to understand.
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 RTT mappings are typically thought of in terms of their ''rows''. This mapping <math>M</math> has two rows; following mathematical conventions, let's call them <math>𝒎_1</math> and <math>𝒎_2</math>. And so to notate this mapping in EBK, we can first imagine capturing the rows as bras like we would normally: <math>𝒎_1</math> = {{map|1 0 -4}} and <math>𝒎_2</math> = {{map|0 1 4}}. Then, to put them together, we can think of this matrix as a single column containing these two rows, or in other words, a ket containing the two bras: {{ket|<math>𝒎_1</math> <math>𝒎_2</math>}}, or fully written out, {{ket|{{map|1 0 -4}} {{map|0 1 4}}}}.
-For another example, the canonical [[comma basis]] for 7-ET consists of the two commas 2187/2048 and 135/128, with PC-vectors {{vector|-11 7}} and {{vector|-7 3 1}}, respectively. As a matrix <math>C</math>, we'd see this as:
+For another example, the canonical [[comma basis]] for 7-ET consists of the two commas 2187/2048 and 135/128, with PC-vectors {{vector|-11 7}} and {{vector|-7 3 1}}, respectively. As a matrix <math>\mathrm{C}</math>, we'd see this as: