Show that each of the following matrices is orthogonal and...

Show that each of the following matrices is orthogonal and find the rotation and/or reflection it produces as an operator acting on vectors. If a rotation, find the axis and angle; if a reflection, find the reflecting plane and the rotation, if any, about the normal to that plane. $$\frac{1}{3}\left(\begin{array}{rrr} -1 & 2 & 2 \\ 2 & -1 & 2 \\ 2 & 2 & -1 \end{array}\right)$$

Show that each of the following matrices is orthogonal and find the rotation and/or reflection it produces as an operator acting on vectors. If a rotation, find the axis and angle;
if a reflection, find the reflecting plane and the rotation, if any, about the normal to that plane.
$$\frac{1}{3}\left(\begin{array}{rrr}
-1 & 2 & 2 \\
2 & -1 & 2 \\
2 & 2 & -1
\end{array}\right)$$

Key Concepts

Orthogonal Matrices

An orthogonal matrix is a square matrix whose rows and columns are orthonormal vectors, meaning that the matrix multiplied by its transpose yields the identity matrix. This property ensures that the associated linear transformation preserves lengths and angles, making it crucial in applications involving rigid motions and preserving the geometry of vector spaces.

Rotation Matrices

Rotation matrices represent linear transformations that rotate vectors by a fixed angle around a particular axis (in three dimensions) while preserving distances. In the context of orthogonal matrices with determinant 1, they belong to the special orthogonal group and are characterized by having one eigenvalue equal to 1 (corresponding to the invariant axis) and a pair of complex conjugate eigenvalues on the unit circle that encode the rotation angle.

Reflection Matrices

Reflection matrices are orthogonal matrices with determinant -1, representing the reflection of vectors through a plane (or hyperplane in higher dimensions). Such transformations reverse the orientation of the space and are characterized by having one eigenvalue equal to 1 (corresponding to the normal direction of the reflecting plane) and the other eigenvalues depend on the behavior within the reflecting plane, often leading to associated rotations if combined with additional transformations.

Eigenvalues and Eigenvectors

The eigenvalues and eigenvectors of a transformation provide essential insights into its geometric behavior. For orthogonal matrices, eigenvalues lie on the unit circle in the complex plane. Real eigenvalues (particularly 1 or -1) indicate fixed directions under the transformation. In rotations, the eigenvector corresponding to the eigenvalue 1 gives the rotation axis, while in reflections, the eigenvalue 1 is associated with the normal to the reflecting plane, with the remaining eigenvalues describing the action within that plane.

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