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| #REDIRECT [[Linear algebra formalism]] | | #REDIRECT [[Linear algebra formalism]] |
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| == Dot product ==
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| A vector is a list of numbers, written like so: <math> \begin{pmatrix}
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| -2\\
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| 0\\
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| 1
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| \end{pmatrix} </math>.
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| The dot product is a way to combine two vectors to get out a single number. Say we want to take the dot product of the vectors <math> \begin{pmatrix}12\\19\\28\end{pmatrix} </math> and <math> \begin{pmatrix}-2\\0\\1\end{pmatrix} </math>. To do so, follow these steps:
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| * Write the vectors separated by a dot to denote the dot product: <math>
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| \begin{pmatrix}
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| 12\\
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| 19\\
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| 28\\
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| \end{pmatrix}
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| \cdot
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| \begin{pmatrix}
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| -2\\
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| 0\\
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| 1
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| \end{pmatrix}
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| </math>
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| ** This may also be notated <math> \langle 12, 19, 28 \vert -2, 0, 1\rangle </math>; from this derives the notation for vals and monzos.
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| * Multiply the corresponding elements, and add the results together: <math> \left(12\cdot-2\right)+\left(19\cdot0\right)+\left(28\cdot1\right) = -24 + 0 + 28 = 4 </math>
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| == Multiply matrix by vector ==
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| A matrix is a grid of numbers, written like so:
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| <math>
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| \begin{bmatrix}
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| 1 & 0 & -4\\
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| 0 & 1 & 4
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| \end{bmatrix}
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| </math>
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| This matrix can be thought of as a "function" that you apply to a vector to get out another vector. This matrix has 3 columns, meaning the vector it takes as an "input" will have 3 elements, and it has 2 rows, meaning the vector you get out will have 2 elements. So, this is a "function" down from 3-dimensional space to 2-dimensional space.
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| We write this "application" of a matrix like so:
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| <math>
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| \begin{bmatrix}
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| 1 & 0 & -4\\
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| 0 & 1 & 4
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| \end{bmatrix}
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| \begin{bmatrix}
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| -2\\
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| 0\\
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| 1
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| \end{bmatrix}
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| </math>
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| where the second object is the vector.
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| To write the first element of our output, we take the dot product of the first row of our matrix with our vector: <math>\begin{pmatrix}
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| 1\\
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| 0\\
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| -4\\
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| \end{pmatrix}
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| \cdot
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| \begin{pmatrix}
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| -2\\
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| 0\\
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| 1
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| \end{pmatrix}
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| = \left(1\cdot-2\right)+\left(0\cdot0\right)+\left(-4\cdot1\right) = -2 + 0 + -4 = -6 </math>
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| We do the same thing for the second element of our output, computing <math>
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| \begin{pmatrix}
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| 0\\
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| 1\\
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| 4\\
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| \end{pmatrix}
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| \cdot
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| \begin{pmatrix}
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| -2\\
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| 0\\
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| 1
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| \end{pmatrix}
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| = 4
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| </math>.
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| Thus, our output is <math> \begin{bmatrix}
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| -6\\
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| 4\\
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| \end{bmatrix}</math> .
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|
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| == Multiply matrix by matrix ==
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| A matrix can act on another matrix, as well. In this case, the matrix on the right can simply be treated as several vectors next to each other.
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|
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| <math>
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| \begin{bmatrix}
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| 1 & 0 & -4\\
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| 0 & 1 & 4
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| \end{bmatrix}
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| \begin{bmatrix}
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| 1 & -1 & -2\\
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| 0 & 1 & 0\\
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| 0 & 0 & 1
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 1 & -1 & -6\\
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| 0 & 1 & 4
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| \end{bmatrix}
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| </math>
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| == Multiply row vector by matrix ==
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| This is to taking the dot product as matrix*matrix multiplication is to matrix*vector multiplication. You take the dot product of the row vector with each successive column of the matrix, and write the result as another row vector. Any matrix*vector operation can be rewritten in this format by swapping rows and columns; the reason these are distinguished is because it is conventional to represent certain things as column vectors and different things as row vectors (i.e. monzos and vals); in this case, vectors represented as rows are called "covectors".
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| == Determinant ==
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| todo
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| == Practical usage ==
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| Let's say we want to determine the tuning of 6/5 in quarter-comma meantone.
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| === Stage 1: JI to temperament ===
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| <math>
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| \begin{bmatrix}
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| 1 & 0 & -4\\
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| 0 & 1 & 4
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| \end{bmatrix}
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| </math>
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| This matrix I've been using as an example is actually a "function" that converts a 5-limit interval in monzo format (with the entries corresponding to powers of 2, 3, and 5) into a corresponding meantone interval in an analogous format (with the entries representing powers of meantone's tempered 2 and 3) called "tempered monzos" or "tmonzos".
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| So, let's take the monzo for 6/5, [1 1 -1⟩, and apply this matrix to it: <math>
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| \begin{bmatrix}
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| 1 & 0 & -4\\
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| 0 & 1 & 4
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| \end{bmatrix}
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| \begin{bmatrix}
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| 1\\
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| 1\\
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| -1
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 5\\
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| -3
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| \end{bmatrix}</math>.
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| The result is the meantone tmonzo representing a tempered 6/5.
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| === Stage 2: Temperament to tuning ===
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| Now, we have the tmonzo. We'll be introducing something called a tval, which gives us a specific tuning of our temperament the same way a regular val gives us a specific tuning of just intonation. The quarter-comma meantone tval for this meantone mapping is ⟨1200 ~1896.5784] in cents. This is where the dot product comes in: <math> \langle 1200, ~1896.5784 \vert 5, -3\rangle </math>.
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| Computing this dot product yields ~310.265, which is exactly the size of the QCM minor third in cents!
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| [[Category:Math]][[Category:Guides]]
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