Lines of invariance question

  • Thread starter morry
  • Start date
  • #1
136
0
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.
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
41,833
956
"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.
 
  • #3
136
0
Thanks for the help. Makes things clearer now. :)
 

Related Threads on Lines of invariance question

Replies
3
Views
2K
  • Last Post
Replies
1
Views
1K
Replies
2
Views
1K
Replies
3
Views
1K
  • Last Post
Replies
8
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
6
Views
5K
  • Last Post
Replies
3
Views
3K
Top