91Ó°ÊÓ

Eigenvalues are the special set of scalar values that is associated with the set of linear equations most probably in the matrix equations. The eigenvectors are also termed as characteristic roots.

Counterclockwise rotation about the ${\stackrel{→}{e}}_3$axis through an angle of ${90}^{°}$.

Objective is to determine all the eigenvalues and eigenvectors of this transformation. Also check the diagonalizability of T by calculating an Eigen basis.

Under the transformation T, any point lie on the ${\stackrel{→}{e}}_3$-axis will get reflect to itself. Or one can say that any vector $\stackrel{→}{v}$that is collinear to ${\stackrel{→}{e}}_3$-axis will remain unchanged. Mathematically, role="math" localid="1659527061971" $Tv=\stackrel{→}{v}$

The eigenvalues of transformation $Tv=\stackrel{→}{v}$will be 1 only because $(T-l)\stackrel{→}{v}=0$. Note that corresponding eigenvector will be $\stackrel{→}{v}={\stackrel{→}{v}}_3$(along the ${\stackrel{→}{e}}_3$-axis).

If the vector $\stackrel{→}{v}$reflects into a vector that is not collinear with ${\stackrel{→}{e}}_3$, then there does not exist any eigenvalue. In this case Eigen basis is also not possible.