Thread: Linear Algebra- Normal and Self-Adjoint Operators

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!!!

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?

Originally Posted by Opalg

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?

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 orthogonal basis is just a basis where if you take the inner product of any 2 different elements it will always be 0.

an orthonormal basis is the same but all vectors in the basis have magnitude 1

Originally Posted by mathgirl

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.

okay...thank you!!!