Angle preserving linear transformations

[bigli]

If x,y of R^n (as a normed vector space) are non-zero, the angle between x and y, denoted
<(x,y), is defined as arccos x.y/(|x||y|).
The linear transformation T :R^n----->R^n
is angle preserving if T is 1-1, and for x,y of R^n (x,y are non zero) we have
<(Tx,Ty) = <(x,y).

what are all angle preserving transformations T :R^N---->R^N ?

I guess that this quastion is connected with eigenvalues of T.please help me!

 

[matt grime]

The question is not (really) about eigenvalues. It is about geometry. You need to visualize what angle preserving means. Start with the plane, and R^3 (since it is not possible to visualize higher dimensions really - you must do it by analogy).

 

[tacman]

An angle preserving linear transformation is a linear transformation that preserves the angle between two vectors. This means that the angle between the transformed vectors is the same as the angle between the original vectors. In other words, if <(x,y) is the angle between two vectors x and y, then <(Tx,Ty) = <(x,y).

To find all angle preserving transformations T :R^n---->R^n, we can look at the eigenvalues of T. If T has distinct eigenvalues, then it is not an angle preserving transformation. This is because the angle between two eigenvectors will not be preserved after the transformation. However, if T has repeated eigenvalues, then it is an angle preserving transformation. This is because repeated eigenvalues indicate that there is a subspace that is invariant under T, and the angle between vectors in this subspace will be preserved after the transformation.

For example, consider the transformation T :R^2---->R^2 given by T(x,y) = (x+y, x+y). This transformation has the eigenvalue 2 with multiplicity 2, indicating that there is a subspace (the line y = x) that is invariant under T. Therefore, T is an angle preserving transformation.

In general, any transformation that has a diagonalizable matrix representation with repeated eigenvalues will be an angle preserving transformation. This includes transformations such as rotations and reflections, as well as scaling transformations along a subspace.

In summary, the set of all angle preserving transformations T :R^n---->R^n includes transformations with repeated eigenvalues, such as rotations, reflections, and scaling transformations along a subspace.