You can express T by means of a matrix: . To find the eigenvectors, you first need to find the eigenvalues, which are the solutions of . Can you take it from there?
Let T be a linear operator on an inner product space V. Determine whether T is normal, self-adjoint, or neither. If possible, produce an orthonormal basis of eigenvectors of T for V and list the corresponding eigenvalues.
V= R^2 and T is defined by
T(a,b) = (2a-2b, -2a+5b)
Okay, so I already know that T is self-adjoint because T=T*
I just don't know how to do the second part for this particular problem.
Thanks so much!!!
ahh~~ duh! thank you so much!
I guess when it said "orthonormal basis" I froze and completely didn't know how to do it. So, how do you know if it is an orthonormal basis? what is the difference between that and other basis? I read the definition on my book but I still cannot see it clearly. Thanks again!
An orthonormal basis is one whose elements are mutually orthogonal and all have norm 1.
There's a theorem which says that if T is selfadjoint then eigenvectors from distinct eigenvalues are always orthogonal, so you get that property for free. To ensure that the eigenvectors have norm 1, just multiply them by appropriate scalars.