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

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

Key Concepts

Eigenvalue Analysis

Eigenvalue and eigenvector analysis is essential in understanding the geometric action of a transformation matrix. For rotations, the invariant axis is the eigenvector corresponding to the eigenvalue 1, whereas in reflections, the plane of reflection is spanned by the eigenvectors with eigenvalue 1 and the normal to the plane corresponds to the eigenvalue -1. This approach helps to identify and separate the rotational and reflective components of a transformation.

Reflection Matrix

A reflection matrix is another form of orthogonal matrix that represents a reflection across a plane (or hyperplane) in space. Unlike rotations, reflection matrices have a determinant of -1. The reflection plane is determined by the set of all vectors that are unchanged under the transformation (eigenvalue 1), while the vector perpendicular to the plane is reversed in direction (eigenvalue -1).

Determinant Considerations

The determinant of an orthogonal matrix aids in distinguishing between rotations and reflections. A determinant of 1 indicates a proper rotation, while a determinant of -1 signals the presence of a reflection, or an improper rotation if combined with another rotational element. This concept is critical when classifying and understanding the nature of geometric transformations.

Orthogonal Matrix

An orthogonal matrix is defined by the property that its transpose is equal to its inverse, meaning that it preserves dot products, lengths, and angles. This property implies that transformations represented by orthogonal matrices are isometries in Euclidean space. Checking that a matrix is orthogonal involves verifying that multiplying the matrix by its transpose yields the identity matrix.

Rotation Matrix

A rotation matrix is a special type of orthogonal matrix that represents a rotation about a fixed axis by a specific angle, without altering the distances or angles between vectors. In three dimensions, finding the axis of rotation typically involves identifying the eigenvector corresponding to the eigenvalue 1, and the rotation angle can be computed using the trace of the matrix. Such matrices have determinant 1, distinguishing them from reflections.

Please give Ace some feedback