Question

Question: In Exercises 31 and 32, let A be the matrix of the linear transformation T. Without writing A, find an eigenvalue of A and describe the Eigenspace.

32. T is the transformation on \({\mathbb{R}^3}\) that rotates points about some line through the origin.

Short answer

The Eigenvalue is \(1\) , and the Eigenspace is the axis of rotation.

Step-by-step solution

Given information

Here, \(T\) is the transformation on \({\mathbb{R}^3}\) and \(T\) rotates some points about some line through the origin.

Find Eigenvalue and Eigenspace of A

Consider \(l\) be the axis of rotation, now observe that any vector on this line will not change by this rotation, that is, \(T{\bf{v}} = {\bf{v}}\).

This implies that \(l\) is an eigenspace corresponding to the Eigenvalue \(\lambda = 1\).

Now, since the rotation does not change the vector's length, there may be one additional real Eigenvalue \(\lambda = - 1\).

Note that if this Eigenvalue exists, then this implies that after the rotation, the vector \({\bf{v}}\)becomes \( - {\bf{v}}\). This is possible only for rotation by \(180^\circ \)of any vector which is perpendicular to the axis \(l\).

Hence,If the rotation is by \(\alpha = 180^\circ \), then there is only one real Eigenvalue \(\lambda = 1\) , and the Eigenspace is the axis of rotation.

If, the rotation is by \(\alpha \ne 180^\circ \), then there is an additional real Eigenvalue \(\lambda = - 1\) , and its Eigenspace is the plane that passes through the origin and also perpendicular to the axis of rotation.