Mathematical guide/Matrix operations - Revision history

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			#REDIRECT [[Linear algebra formalism]]

	== Dot product ==		== Dot product ==
	A vector is a list of numbers, written like so: <math> \begin{pmatrix}		A vector is a list of numbers, written like so: <math> \begin{pmatrix}

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	Computing this dot product yields ~310.265, which is exactly the size of the QCM minor third in cents!		Computing this dot product yields ~310.265, which is exactly the size of the QCM minor third in cents!

			[[Category:Math]][[Category:Guides]]

VectorGraphics at 20:42, 14 April 2025

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	== Multiply row vector by matrix ==		== Multiply row vector by matrix ==
			This is to taking the dot product as matrixmatrix multiplication is to matrixvector 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".

	== Determinant ==		== Determinant ==

VectorGraphics at 06:32, 14 April 2025

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	\end{bmatrix}		\end{bmatrix}
	</math>		</math>

			== Multiply row vector by matrix ==

	== Determinant ==		== Determinant ==

VectorGraphics at 05:49, 14 April 2025

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@@ Line 6: / Line 6: @@
 \end{pmatrix} </math>.
-The dot product is a way to combine two vectors to get out a single number.
+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:
 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>.
 * Write the vectors separated by a dot to denote the dot product: <math>
 \begin{pmatrix}
@@ Line 21: / Line 20: @@
 \end{pmatrix}
 </math>
-**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.
+** 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.
 * 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>
@@ Line 53: / Line 52: @@
 where the second object is the vector.
-To write the first element of our output, we take the dot product of the first row of our matrix with our vector:
+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}
 \\
 \\
@@ Line 85: / Line 83: @@
 -6\\
 \\
-\end{bmatrix}</math>
+\end{bmatrix}</math> .
 == Multiply matrix by matrix ==
 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.
 <math>
@@ Line 109: / Line 104: @@
 \end{bmatrix}
 </math>

VectorGraphics at 03:48, 14 April 2025

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← Older revision		Revision as of 03:48, 14 April 2025
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	~~Todo~~		== Dot product ==
			A vector is a list of numbers, written like so: <math> \begin{pmatrix}
			-2\\
			0\\
			1
			\end{pmatrix} </math>.

			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:
			* Write the vectors separated by a dot to denote the dot product: <math>
			\begin{pmatrix}
			12\\
			19\\
			28\\
			\end{pmatrix}
			\cdot
			\begin{pmatrix}
			-2\\
			0\\
			1
			\end{pmatrix}
			</math>
			**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.
			* 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>

			== Multiply matrix by vector ==
			A matrix is a grid of numbers, written like so:

			<math>
			\begin{bmatrix}
			1 & 0 & -4\\
			0 & 1 & 4
			\end{bmatrix}
			</math>

			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.

			We write this "application" of a matrix like so:

			<math>
			\begin{bmatrix}
			1 & 0 & -4\\
			0 & 1 & 4
			\end{bmatrix}
			\begin{bmatrix}
			-2\\
			0\\
			1
			\end{bmatrix}
			</math>

			where the second object is the vector.

			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}
			1\\
			0\\
			-4\\
			\end{pmatrix}
			\cdot
			\begin{pmatrix}
			-2\\
			0\\
			1
			\end{pmatrix}
			= \left(1\cdot-2\right)+\left(0\cdot0\right)+\left(-4\cdot1\right) = -2 + 0 + -4 = -6 </math>

			We do the same thing for the second element of our output, computing <math>
			\begin{pmatrix}
			0\\
			1\\
			4\\
			\end{pmatrix}
			\cdot
			\begin{pmatrix}
			-2\\
			0\\
			1
			\end{pmatrix}
			= 4
			</math>.

			Thus, our output is <math> \begin{bmatrix}
			-6\\
			4\\
			\end{bmatrix}</math>
			.

			== Multiply matrix by matrix ==

			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.


			<math>
			\begin{bmatrix}
			1 & 0 & -4\\
			0 & 1 & 4
			\end{bmatrix}
			\begin{bmatrix}
			1 & -1 & -2\\
			0 & 1 & 0\\
			0 & 0 & 1
			\end{bmatrix}
			=
			\begin{bmatrix}
			1 & -1 & -6\\
			0 & 1 & 4
			\end{bmatrix}
			</math>

VectorGraphics: VectorGraphics moved page Mathematical guide/Matrix multiplication to Mathematical guide/Matrix operations

2025-04-14T03:00:05Z

VectorGraphics moved page Mathematical guide/Matrix multiplication to Mathematical guide/Matrix operations

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