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. $$\frac{1}{2}\left(\begin{array}{ccc} 1 & \sqrt{2} & -1 \\ \sqrt{2} & 0 & \sqrt{2} \\ 1 & -\sqrt{2} & -1 \end{array}\right)$$

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.
$$\frac{1}{2}\left(\begin{array}{ccc}
1 & \sqrt{2} & -1 \\
\sqrt{2} & 0 & \sqrt{2} \\
1 & -\sqrt{2} & -1
\end{array}\right)$$

Key Concepts

Orthogonal Matrix

An orthogonal matrix is one whose transpose is equal to its inverse, meaning Q^T Q = I. This property ensures that the transformation preserves lengths and angles, making it ideal for representing rotations and/or reflections in Euclidean space. The orthogonality also implies that the columns (and rows) of the matrix form an orthonormal basis.

Rotation Operator in 3D

In three dimensions, a rotation operator is represented by an orthogonal matrix with determinant equal to 1. Such a transformation rotates vectors about an axis while preserving distances. The axis of rotation corresponds to the eigenvector associated with the eigenvalue 1, and the rotation angle can be computed from the other eigenvalues or from the trace of the rotation matrix.

Reflection Operator in 3D

A reflection operator in three dimensions is given by an orthogonal matrix with determinant -1. This matrix reflects vectors across a plane while preserving lengths. The invariant direction in this case, typically corresponding to the normal of the reflecting plane, is determined by the eigenvector associated with the eigenvalue -1, with the rest of the space being flipped about the plane orthogonal to this direction.

Eigenvalues and Eigenvectors in Transformation Analysis

Eigenvalues and eigenvectors play a crucial role in understanding the geometric action of linear transformations such as rotations and reflections. In the context of 3D transformations, identifying an eigenvector with eigenvalue 1 (for rotations) or -1 (for reflections) helps locate the rotation axis or the normal to the reflecting plane. The remaining eigenvalues, which are often complex in the rotation case, encode the angle of rotation.

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