Matrices that move the plane
Tonight's idea
A \(2 \times 2\) matrix is a rule for moving every point of the plane at once:
\(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} ax + by \\ cx + dy \end{pmatrix}\)
Read a matrix by its columns. Feed in \((1, 0)\) and out comes \((a, c)\), the first column; feed in \((0, 1)\) and out comes \((b, d)\). Every point is \(x\) lots of \((1, 0)\) plus \(y\) lots of \((0, 1)\), so the whole plane follows: the unit square becomes the parallelogram spanned by the columns, grid lines stay straight and parallel, and the origin never moves.
Rotation by \(\theta\) sends \((1, 0)\) to \((\cos\theta, \sin\theta)\) and \((0, 1)\) to \((-\sin\theta, \cos\theta)\), so \(R_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\). A shear \(\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}\) tips \((0, 1)\) over to \((k, 1)\).
The determinant \(ad - bc\) is the signed area (negative if the square has been flipped over) of the image of the unit square. Zero means the square has been squashed flat onto a line (or a single point) and no inverse exists. Otherwise:
\(\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \dfrac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\)
Two equations are one matrix equation: \(2x + y = 7,\ x + 3y = 11\) is \(A\binom{x}{y} = \binom{7}{11}\) with \(A = \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix}\), so \(\binom{x}{y} = A^{-1}\binom{7}{11}\). Doing one matrix after another is the product, and \(MN\) means "do \(N\), then \(M\)". Order matters. Demo: mathdetours.com/demos/matrix.html.
Watch one: a rotation and a shear, both orders
\(R_{90°} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\) sends \((3, 1)\) to \((-1, 3)\), a quarter turn anticlockwise. With \(S = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}\), rotate then shear is \(SR = \begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}\), but shear then rotate is \(RS = \begin{pmatrix} 0 & -1 \\ 1 & 1 \end{pmatrix}\).
Your turn
- Write the matrix for each using the column rule, then apply it to \((2, 5)\): (a) reflection in the \(x\)-axis; (b) reflection in the line \(y = x\); (c) rotation by \(180°\); (d) stretching by a factor of 3 horizontally only.
- Write \(R_{60°}\) exactly and apply it to \((2, 0)\). Explain without computing why \(R_{60°}^6 = I\).
- Find the determinants of \(R_\theta\), of \(\begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix}\), and of \(\begin{pmatrix} 2 & 4 \\ 1 & 2 \end{pmatrix}\). For the last one, describe the line the whole plane is squashed onto.
- Find the inverse of \(A = \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix}\) and use it to solve \(2x + y = 7,\ x + 3y = 11\). Check in the original equations.
Stretch: the directions a matrix cannot turn
Apply \(B = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}\) to \((1, 1)\) and to \((1, -1)\). Each comes back pointing the same way, only stretched. Read off the two stretch factors, multiply them, and compare with \(\det B\). Why is that no coincidence?
Answers
- (a) \(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\): \((2, -5)\). (b) \(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\): \((5, 2)\). (c) \(\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}\): \((-2, -5)\). (d) \(\begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}\): \((6, 5)\).
- \(R_{60°} = \begin{pmatrix} \tfrac12 & -\tfrac{\sqrt3}{2} \\ \tfrac{\sqrt3}{2} & \tfrac12 \end{pmatrix}\); \((2, 0) \mapsto (1, \sqrt3)\). Six turns of \(60°\) make a full turn, which moves nothing.
- \(\cos^2\theta + \sin^2\theta = 1\): areas unchanged. \(6 - 1 = 5\): the unit square becomes a parallelogram of area 5. \(4 - 4 = 0\): the columns \((2, 1)\) and \((4, 2)\) point the same way, so everything lands on the line \(y = \tfrac{x}{2}\).
- \(A^{-1} = \tfrac15\begin{pmatrix} 3 & -1 \\ -1 & 2 \end{pmatrix}\). Solution \(\tfrac15(21 - 11,\ -7 + 22) = (2, 3)\). Check: \(4 + 3 = 7\), \(2 + 9 = 11\).
Stretch. \(B(1, 1) = (3, 3)\): factor 3. \(B(1, -1) = (1, -1)\): factor 1. Product \(3 = \det B = 4 - 1\). A square with sides along those two directions is stretched by 3 one way and 1 the other, so its area is multiplied by 3, and the determinant is exactly the area multiplier. These directions are called eigenvectors.