Understanding Lines of Invariance and Their Relationship to Rotational Matrices

[morry]

Just need a couple of things confirmed for me guys.

Firstly, lines of invariance are always real eigenvectors right?

Secondly, how is this line of invariance related to rotational matrices? My line of invariance happens to be the same axis. Finally, how is the angle of the rotation calculated?

Cheers.

 

[HallsofIvy]

"Lines of invariance" ? Are you talking about transformations on some Rⁿ so that "lines of invariance" are lines that are not changed by the transformation?

To be "invariant", either every point on the line is mapped into itself or every point on the line is mapped into another point on the line. In either case, if v is a unit vector in the direction of the line, Av= αv for some real number α. Yes, lines of invariance correspond to real eigen-values. Obviously the lines themselves are neither eigenvalues nor eigen vectors.

Certainly, a rotation around a given axis leaves that axis invariant- the axis is a "line of invariance". As for "how is the angle of the rotation calculated?", that depends on what information you are given. If you are given the rotation matrix, find the eigenvalues. One, with eigenvector in the direction of the axis of rotation, will be 1, the others will be complex conjugates of the form [itex]e^{i\theta}[/itex], where θ is the angle of rotation.

 

[morry]

Thanks for the help. Makes things clearer now. :)