Is a reflection matrix a rotation matrix?

QuestionsCategory: QuestionsIs a reflection matrix a rotation matrix?
mrtester asked 3 years ago

I wonder if a reflection matrix is a rotation as well?

Nasreenkanwal replied 3 years ago

How to sum one matrix into other matrix? [Unrelated link removed by moderator, see posting rules.]

1 Answers
Tomas Akenine-Möller Staff answered 3 years ago

A reflection matrix is usually not considered to be a rotation matrix, since rotation matrices have $\det\mathbf M=1$ and a reflection matrix can have $\det\mathbf M=-1$. An example of a reflection matrix is
\begin{equation}
\mathbf M =
\begin{pmatrix}
-1 & 0 \\
0 & 1
\end{pmatrix},
\end{equation}
whose determinant is $-1$.
Interestingly, the product of two reflection matrices become a rotation matrix. Assume we have another reflection matrix, i.e.,
\begin{equation}
\mathbf N =
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}
\end{equation}
Their product becomes
\begin{equation}
\mathbf M\mathbf N =
\begin{pmatrix}
-1 & 0 \\
0 & 1
\end{pmatrix}
\begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix}
=
\begin{pmatrix}
-1 & 0 \\
0 & -1
\end{pmatrix},
\end{equation}
and as can be seen $\det(\mathbf M\mathbf N)=1$.
Read more about matrices in Chapter 6 in our book.