Describing Matrix/Transformation by Eigens.

[WWGD]

Hi, All:

Say T:R^n --->R^n is a linear map , and that the associated matrix M has a unique eigenvalue l=1. Is M necessarily a rotation matrix about the eigenspace?

Thanks.

 

[morphism]

No - e.g. consider $\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1\end{smallmatrix}\right)$.

 

[WWGD]

I see. I am trying to show that by applying kri+rj , i.e., adding a multiple of k times row i to

row j one row to another row has the effect of rotating one of the k-planes about the

solution subspace, since this is the only way I can conceive that the operation kri+rj

preserves the solution to the system. Can you see how I else I can show this?

 

[morphism]

Solution subspace of what?

 

[WWGD]

Well, I have a soluble , i.e., non-contradictory (homogeneous) system of linear equations .

The fundamental row operations--exchange rows, add a multiple of one

row to another row-- preserve the solutions to the system. If we look at the

solution S to the system geometrically, this is a subspace, possibly trivial. I'm trying

to show that the operation of adding a multiple of row i to row j has the effect of

rotating the n-planes in the system of equations about the solution- space S.

I think the affine case--for non-homogeneous systems, is similar. I've been using

the fundamental theorem of linear algebra that Bacle had mentioned in a similar

problem, but I still can't prove this.