For linear operator T on V, find an ordered basis beta for V such that?

[T]beta is a diagonal matrix.


V= R^3 and T(a, b, c)= (-4a+ 3b- 6c, 6a- 7b+12c, 6a- 6b+ 11c)

For linear operator T on V, find an ordered basis beta for V such that?
What you are looking for are the eigenvectors of the linear operator T.





http://en.wikipedia.org/wiki/Eigenvalue





My guess is that the matrix representation of T is designed to make finding the eigenvectors simple. But in case you can't see it, there is a classic algorithm for finding the eigenvector with the largest eigenvalue:





http://mathworld.wolfram.com/Eigenvector...





Once you do than, you can reduce the operator from 3-D to 2-D and repeat.





Or, take a look at this site:


http://www-math.mit.edu/18.013A/HTML/cha...





I'm not happy with this one in general, but you might prefer it for this particular problem (if the numbers work out, it is the easiest):


http://cnx.org/content/m12083/latest/





There are other algorithms designed for computers and large arrays, but I don't think you'd be interested in them for this problem.