Mathematical guide/Matrix operations

Dot product

A vector is a list of numbers, written like so: [math]\displaystyle{ \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]\displaystyle{ \begin{pmatrix}12\\19\\28\end{pmatrix} }[/math] and [math]\displaystyle{ \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]\displaystyle{ \begin{pmatrix} 12\\ 19\\ 28\\ \end{pmatrix} \cdot \begin{pmatrix} -2\\ 0\\ 1 \end{pmatrix} }[/math]
  • This may also be notated [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \begin{pmatrix} 0\\ 1\\ 4\\ \end{pmatrix} \cdot \begin{pmatrix} -2\\ 0\\ 1 \end{pmatrix} = 4 }[/math].

Thus, our output is [math]\displaystyle{ \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]\displaystyle{ \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]