Editing Orthogonal matrix (section)

===Lower dimensions===
The simplest orthogonal matrices are the {{nowrap|1 × 1}} matrices [1] and [−1], which we can interpret as the identity and a reflection of the real line across the origin.

The {{nowrap|2 × 2}} matrices have the form
<math display="block">\begin{bmatrix}
p & t\\
q & u
\end{bmatrix},</math>
which orthogonality demands satisfy the three equations
<math display="block">\begin{align}
1 & = p^2+t^2, \\
1 & = q^2+u^2, \\
0 & = pq+tu.
\end{align}</math>

In consideration of the first equation, without loss of generality let {{math|1=''p'' = cos ''θ''}}, {{math|1=''q'' = sin ''θ''}}; then either {{math|1=''t'' = −''q''}}, {{math|1=''u'' = ''p''}} or {{math|1=''t'' = ''q''}}, {{math|1=''u'' = −''p''}}. We can interpret the first case as a rotation by {{mvar|θ}} (where {{math|1=''θ'' = 0}} is the identity), and the second as a reflection across a line at an angle of {{math|{{sfrac|''θ''|2}}}}.

<math display="block">
\begin{bmatrix}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta \\
\end{bmatrix}\text{ (rotation), }\qquad
\begin{bmatrix}
\cos \theta & \sin \theta \\
\sin \theta & -\cos \theta \\
\end{bmatrix}\text{ (reflection)}
</math>

The special case of the reflection matrix with {{math|1=''θ'' = 90°}} generates a reflection about the line at 45° given by {{math|1=''y'' = ''x''}} and therefore exchanges {{mvar|x}} and {{mvar|y}}; it is a [[permutation matrix]], with a single 1 in each column and row (and otherwise 0):
<math display="block">\begin{bmatrix}
0 & 1\\
1 & 0
\end{bmatrix}.</math>

The identity is also a permutation matrix.

A reflection is [[Involutory matrix|its own inverse]], which implies that a reflection matrix is [[symmetric matrix|symmetric]] (equal to its transpose) as well as orthogonal. The product of two rotation matrices is a [[rotation matrix]], and the product of two reflection matrices is also a rotation matrix.