Matrix Reflection of a Figure

Background

Previously, we've seen how to rotate a figure using matrices. This last lesson in this series will cover the next matrix transformation, matrix reflection.

Before we continue, it's important that we review how to represent a function with a matrix. We put the ordered pair vertically in the matrix. Say we're given the points, $(x,y)$, $(a,b)$, and $(1, 2)$. We would represent these points in matrix form like so:

$$\begin{bmatrix} x&a&1 \\ y&b&2 \\ \end{bmatrix}$$

As we can see, the coordinate points are vertically paired in the matrix.

Reflection Across x-axis

If we want to reflect a given matrix, $\begin{bmatrix} a&b \\ c&d \\ \end{bmatrix}$, across the $x$-axis, we multiply the given matrix by $\begin{bmatrix}1&0\\0&-1\end{bmatrix}$.

This is another method, compared to what we've seen previously with function reflections across the x-axis.

Reflection Across y-axis

If we want to reflect a given matrix, $\begin{bmatrix} a&b \\ c&d \\ \end{bmatrix}$, across the $y$-axis, we multiply the given matrix by: $\begin{bmatrix}-1&0\\0&1\end{bmatrix}$.

This is another method, compared to what we've seen previously with function reflections across the y-axis.

Reflection Across $y=x$

If we want to reflect a given matrix, $\begin{bmatrix} a&b \\ c&d \\ \end{bmatrix}$, across the line $y=x$, we multiply the given matrix by: $\begin{bmatrix}0&1\\1&0\end{bmatrix}$.

This is also known as finding the inverse. We previously learned how to find the inverse graphically and algebraically.

Reflection Across $y=-x$

If we want to reflect a given matrix, $\begin{bmatrix} a&b \\ c&d \\ \end{bmatrix}$, across the line $y=-x$, we multiply the given matrix by: $\begin{bmatrix}0&-1\\-1&0\end{bmatrix}$.

(Video) Matrix Reflections

by statslectures

This video by statslectures talks about matrix reflections.

Summary

To use matrices to reflect points, we use matrix multiplication! To reflect a figure is to flip it across its line of symmetry.

To flip a figure over the $x$-axis, multiply by: $\begin{bmatrix} 1&0 \\ 0&-1 \end{bmatrix}$

To flip a figure over the $y$-axis, multiply by: $\begin{bmatrix} -1&0 \\ 0&1 \\ \end{bmatrix}$

To flip a figure over the line $y=x$, multiply by: $\begin{bmatrix} 0&1 \\ 1&0 \\ \end{bmatrix}$

Given a triangle with the following three points, $(1,7)$, $(1,2)$, and $(6,2)$, we can write a vector matrix:

$\begin{bmatrix} 1&1&6 \\ 7&2&2 \\ \end{bmatrix}$

If we want to reflect this triangle across the $x$-axis, we will multiply:

$$\begin{bmatrix} 1&0 \\ 0&-1 \\ \end{bmatrix} \begin{bmatrix} 1&1&6 \\ 7&2&2 \\ \end{bmatrix} = \begin{bmatrix} 1&1&6 \\ 7&-2&-2 \\ \end{bmatrix}$$

In this set of lessons, we will learn about matrix rotation, matrix reflection (this lesson), matrix dilation, and matrix translation.

Matrix Reflection of a Figure

If we want to reflect a $2x2$ matrix, we can multiply it by one of the following matrices to get the desired reflection.

These are the transformation matrices:

If we want to reflect a matrix over the $x$-axis, we multiply the matrix by: $\begin{bmatrix} 1&0 \\ 0&-1 \\ \end{bmatrix}$

If we want to reflect a matrix over the $y$-axis, we multiply the matrix by: $\begin{bmatrix} -1&0 \\ 0&1 \\ \end{bmatrix}$

If we want to reflect a matrix over the line $y=x$, we multiply the matrix by: $\begin{bmatrix} 0&1 \\ 1&0 \\ \end{bmatrix}$

If we want to reflect a matrix over the line$y=-x$, we multiply the matrix by: $\begin{bmatrix} 0&-1 \\ -1&0 \\ \end{bmatrix}$

If we plot the points found in the resulting matrix, we will find a reflected image.