Orthonormal Matrices

In The Elements, Euclid considers two figures to be the same if they have the same size and shape. That is, while the triangles below are not equal because they are not the same set of points, they are, for Euclid’s purposes, essentially indistinguishable because we can imagine picking the plane up, sliding it over and rotating it a bit, although not warping or stretching it, and then putting it back down, to superimpose the first figure on the second. (Euclid never explicitly states this principle but he uses it often [Casey].)

In modern terms “picking the plane up …” is taking a map from the plane to itself. Euclid considers only transformations that may slide or turn the plane but not bend or stretch it. Accordingly, define a map $f:{\mathrm{ℝ}}^2\to {\mathrm{ℝ}}^2$ to be distance-preserving or a rigid motion or an isometry if for all points $P_1,P_2\in {\mathrm{ℝ}}^2$, the distance from $f(P_1)$ to $f(P_2)$ equals the distance from $P_1$ to $P_2$. We also define a plane figure to be a set of points in the plane and we say that two figures are congruent if there is a distance-preserving map from the plane to itself that carries one figure onto the other.

Many statements from Euclidean geometry follow easily from these definitions. Some are: (i) collinearity is invariant under any distance-preserving map (that is, if $P_1$, $P_2$, and $P_3$ are collinear then so are $f(P_1)$, $f(P_2)$, and $f(P_3)$), (ii) betweeness is invariant under any distance-preserving map (if $P_2$ is between $P_1$ and $P_3$ then so is $f(P_2)$ between $f(P_1)$ and $f(P_3)$), (iii) the property of being a triangle is invariant under any distance-preserving map (if a figure is a triangle then the image of that figure is also a triangle), (iv) and the property of being a circle is invariant under any distance-preserving map. In 1872, F. Klein suggested that we can define Euclidean geometry as the study of properties that are invariant under these maps. (This forms part of Klein’s Erlanger Program, which proposes the organizing principle that we can describe each kind of geometry—Euclidean, projective, etc.— as the study of the properties that are invariant under some group of transformations. The word ‘group’ here means more than just ‘collection’ but that lies outside of our scope.)

We can use linear algebra to characterize the distance-preserving maps of the plane.

To begin, observe that there are distance-preserving transformations of the plane that are not linear. The obvious example is this translation.

$$\left(\begin{array}{c}x\\ y\end{array}\right)\mapsto \left(\begin{array}{c}x\\ y\end{array}\right)+\left(\begin{array}{c}1\\ 0\end{array}\right)=\left(\begin{array}{c}x+1\\ y\end{array}\right)$$

However, this example turns out to be the only one, in that if $f$ is distance-preserving and sends $\overrightarrow{0}$ to ${\overrightarrow{v}}_0$ then the map $\overrightarrow{v}\mapsto f(\overrightarrow{v})-{\overrightarrow{v}}_0$ is linear. That will follow immediately from this statement: a map $t$ that is distance-preserving and sends $\overrightarrow{0}$ to itself is linear. To prove this equivalent statement, consider the standard basis and suppose that

$$t({\overrightarrow{e}}_1)=\left(\begin{array}{c}a\\ b\end{array}\right)t({\overrightarrow{e}}_2)=\left(\begin{array}{c}c\\ d\end{array}\right)$$

for some $a,b,c,d\in \mathrm{ℝ}$. To show that $t$ is linear we can show that it can be represented by a matrix, that is, that $t$ acts in this way for all $x,y\in \mathrm{ℝ}$.

$$\begin{array}{cc}\overrightarrow{v}=\left(\begin{array}{c}x\\ y\end{array}\right)\stackrel{t}{⟼}\left(\begin{array}{c}ax+cy\\ bx+dy\end{array}\right)& \text{(}\ast \text{)}\end{array}$$

Recall that if we fix three non-collinear points then we can determine any point by giving its distance from those three. So we can determine any point $\overrightarrow{v}$ in the domain by its distance from $\overrightarrow{0}$, ${\overrightarrow{e}}_1$, and ${\overrightarrow{e}}_2$. Similarly, we can determine any point $t(\overrightarrow{v})$ in the codomain by its distance from the three fixed points $t(\overrightarrow{0})$, $t({\overrightarrow{e}}_1)$, and $t({\overrightarrow{e}}_2)$ (these three are not collinear because, as mentioned above, collinearity is invariant and $\overrightarrow{0}$, ${\overrightarrow{e}}_1$, and ${\overrightarrow{e}}_2$ are not collinear). Because $t$ is distance-preserving we can say more: for the point $\overrightarrow{v}$ in the plane that is determined by being the distance $d_0$ from $\overrightarrow{0}$, the distance $d_1$ from ${\overrightarrow{e}}_1$, and the distance $d_2$ from ${\overrightarrow{e}}_2$, its image $t(\overrightarrow{v})$ must be the unique point in the codomain that is determined by being $d_0$ from $t(\overrightarrow{0})$, $d_1$ from $t({\overrightarrow{e}}_1)$, and $d_2$ from $t({\overrightarrow{e}}_2)$. Because of the uniqueness, checking that the action in ($\ast$) works in the $d_0$, $d_1$, and $d_2$ cases

$$\mathrm{dist}(\left(\begin{array}{c}x\\ y\end{array}\right),\overrightarrow{0})=\mathrm{dist}(t(\left(\begin{array}{c}x\\ y\end{array}\right)),t(\overrightarrow{0}))=\mathrm{dist}(\left(\begin{array}{c}ax+cy\\ bx+dy\end{array}\right),\overrightarrow{0})$$

(we assumed that $t$ maps $\overrightarrow{0}$ to itself)

$$\mathrm{dist}(\left(\begin{array}{c}x\\ y\end{array}\right),{\overrightarrow{e}}_1)=\mathrm{dist}(t(\left(\begin{array}{c}x\\ y\end{array}\right)),t({\overrightarrow{e}}_1))=\mathrm{dist}(\left(\begin{array}{c}ax+cy\\ bx+dy\end{array}\right),\left(\begin{array}{c}a\\ b\end{array}\right))$$

and

$$\mathrm{dist}(\left(\begin{array}{c}x\\ y\end{array}\right),{\overrightarrow{e}}_2)=\mathrm{dist}(t(\left(\begin{array}{c}x\\ y\end{array}\right)),t({\overrightarrow{e}}_2))=\mathrm{dist}(\left(\begin{array}{c}ax+cy\\ bx+dy\end{array}\right),\left(\begin{array}{c}c\\ d\end{array}\right))$$

suffices to show that ($\ast$) describes $t$. Those checks are routine.

Thus any distance-preserving $f:{\mathrm{ℝ}}^2\to {\mathrm{ℝ}}^2$ is a linear map plus a translation, $f(\overrightarrow{v})=t(\overrightarrow{v})+{\overrightarrow{v}}_0$ for some constant vector ${\overrightarrow{v}}_0$ and linear map $t$ that is distance-preserving. So in order to understand distance-preserving maps what remains is to understand distance-preserving linear maps.

Not every linear map is distance-preserving. For example $\overrightarrow{v}\mapsto 2\overrightarrow{v}$ does not preserve distances.

But there is a neat characterization: a linear transformation $t$ of the plane is distance-preserving if and only if both $\Vert t({\overrightarrow{e}}_1)\Vert =\Vert t({\overrightarrow{e}}_2)\Vert =1$, and $t({\overrightarrow{e}}_1)$ is orthogonal to $t({\overrightarrow{e}}_2)$. The ‘only if’ half of that statement is easy—because $t$ is distance-preserving it must preserve the lengths of vectors and because $t$ is distance-preserving the Pythagorean theorem shows that it must preserve orthogonality. To show the ‘if’ half we can check that the map preserves lengths of vectors because then for all $\overrightarrow{p}$ and $\overrightarrow{q}$ the distance between the two is preserved $\Vert t(\overrightarrow{p}-\overrightarrow{q})\Vert =\Vert t(\overrightarrow{p})-t(\overrightarrow{q})\Vert =\Vert \overrightarrow{p}-\overrightarrow{q}\Vert$. For that check let

$$\overrightarrow{v}=\left(\begin{array}{c}x\\ y\end{array}\right)t({\overrightarrow{e}}_1)=\left(\begin{array}{c}a\\ b\end{array}\right)t({\overrightarrow{e}}_2)=\left(\begin{array}{c}c\\ d\end{array}\right)$$

and with the ‘if’ assumptions that $a^2+b^2=c^2+d^2=1$ and $ac+bd=0$ we have this.

$$\begin{array}{rl}\Vert t(\overrightarrow{v}){\Vert }^2& =(ax+cy{)}^2+(bx+dy{)}^2\\ & =a^2{x}^2+2acxy+c^2{y}^2+b^2{x}^2+2bdxy+d^2{y}^2\\ & =x^2(a^2+b^2)+y^2(c^2+d^2)+2xy(ac+bd)\\ & =x^2+y^2\\ & =\Vert \overrightarrow{v}{\Vert }^2\end{array}$$

One thing that is neat about this characterization is that we can easily recognize matrices that represent such a map with respect to the standard bases: the columns are of length one and are mutually orthogonal. This is an orthonormal matrix (or, more informally, orthogonal matrix since people often use this term to mean not just that the columns are orthogonal but also that they have length one).

We can leverage this characterization to understand the geometric actions of distance-preserving maps. Because $\Vert t(\overrightarrow{v})\Vert =\Vert \overrightarrow{v}\Vert$, the map $t$ sends any $\overrightarrow{v}$ somewhere on the circle about the origin that has radius equal to the length of $\overrightarrow{v}$. In particular, ${\overrightarrow{e}}_1$ and ${\overrightarrow{e}}_2$ map to the unit circle. What’s more, once we fix the unit vector ${\overrightarrow{e}}_1$ as mapped to the vector with components $a$ and $b$ then there are only two places where ${\overrightarrow{e}}_2$ can go if its image is to be perpendicular to the first vector’s image: it can map either to one where ${\overrightarrow{e}}_2$ maintains its position a quarter circle clockwise from ${\overrightarrow{e}}_1$

${\mathrm{Rep}}_{{\mathrm{ℰ}}_2,{\mathrm{ℰ}}_2}(t)=\left(\begin{array}{cc}a& -b\\ b& a\end{array}\right)$

or to one where it goes a quarter circle counterclockwise.

${\mathrm{Rep}}_{{\mathrm{ℰ}}_2,{\mathrm{ℰ}}_2}(t)=\left(\begin{array}{cc}a& b\\ b& -a\end{array}\right)$

The geometric description of these two cases is easy. Let $\theta$ be the counterclockwise angle between the $x$-axis and the image of ${\overrightarrow{e}}_1$. The first matrix above represents, with respect to the standard bases, a rotation of the plane by $\theta$ radians.

$\left(\begin{array}{c}x\\ y\end{array}\right)\stackrel{t}{⟼}\left(\begin{array}{c}x\cos \theta -y\sin \theta \\ x\sin \theta +y\cos \theta \end{array}\right)$

The second matrix above represents a reflection of the plane through the line bisecting the angle between ${\overrightarrow{e}}_1$ and $t({\overrightarrow{e}}_1)$.

$\left(\begin{array}{c}x\\ y\end{array}\right)\stackrel{t}{⟼}\left(\begin{array}{c}x\cos \theta +y\sin \theta \\ x\sin \theta -y\cos \theta \end{array}\right)$

(This picture shows ${\overrightarrow{e}}_1$ reflected up into the first quadrant and ${\overrightarrow{e}}_2$ reflected down into the fourth quadrant.)

Note: in the domain the angle between ${\overrightarrow{e}}_1$ and ${\overrightarrow{e}}_2$ runs counterclockwise, and in the first map above the angle from $t({\overrightarrow{e}}_1)$ to $t({\overrightarrow{e}}_2)$ is also counterclockwise, so it preserves the orientation of the angle. But the second map reverses the orientation. A distance-preserving map is direct if it preserves orientations and opposite if it reverses orientation.

With that, we have characterized the Euclidean study of congruence. It considers, for plane figures, the properties that are invariant under combinations of (i) a rotation followed by a translation, or (ii) a reflection followed by a translation (a reflection followed by a non-trivial translation is a glide reflection).

Another idea encountered in elementary geometry, besides congruence of figures, is that figures are similar if they are congruent after a change of scale. The two triangles below are similar since the second is the same shape as the first but $3/2$-ths the size.

From the above work we have that figures are similar if there is an orthonormal matrix $T$ such that the points $\overrightarrow{q}$ on one figure are the images of the points $\overrightarrow{p}$ on the other figure by $\overrightarrow{q}=(kT)\overrightarrow{v}+{\overrightarrow{p}}_0$ for some nonzero real number $k$ and constant vector ${\overrightarrow{p}}_0$.

Although these ideas are from Euclid, mathematics is timeless and they are still in use today. One application of the maps studied above is in computer graphics. We can, for example, animate this top view of a cube by putting together film frames of it rotating; that’s a rigid motion.

Frame 1	Frame 2	Frame 3

We could also make the cube appear to be moving away from us by producing film frames of it shrinking, which gives us figures that are similar.

Frame 1:	Frame 2:	Frame 3:

Computer graphics incorporates techniques from linear algebra in many other ways (see Exercise 4).

A beautiful book that explores some of this area is [Weyl]. More on groups, of transformations and otherwise, is in any book on Modern Algebra, for instance [Birkhoff & MacLane]. More on Klein and the Erlanger Program is in [Yaglom].

References cited in this section