# Linear Algebra- Normal and Self-Adjoint Operators

• Dec 4th 2007, 12:30 AM
mathgirl
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)

I just don't know how to do the second part for this particular problem.
Thanks so much!!!
• Dec 4th 2007, 11:24 AM
Opalg
You can express T by means of a matrix: $\displaystyle T\begin{bmatrix}a\\b\end{bmatrix} = \begin{bmatrix}2&-2\\-2&5\end{bmatrix}\begin{bmatrix}a\\b\end{bmatrix}$. To find the eigenvectors, you first need to find the eigenvalues, which are the solutions of $\displaystyle \begin{vmatrix}2-\lambda&-2\\-2&5-\lambda\end{vmatrix}$. Can you take it from there?
• Dec 4th 2007, 06:45 PM
mathgirl
Quote:

Originally Posted by Opalg
You can express T by means of a matrix: $\displaystyle T\begin{bmatrix}a\\b\end{bmatrix} = \begin{bmatrix}2&-2\\-2&5\end{bmatrix}\begin{bmatrix}a\\b\end{bmatrix}$. To find the eigenvectors, you first need to find the eigenvalues, which are the solutions of $\displaystyle \begin{vmatrix}2-\lambda&-2\\-2&5-\lambda\end{vmatrix}$. 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!
• Dec 4th 2007, 07:49 PM
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
• Dec 5th 2007, 12:12 AM
Opalg
Quote:

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.
• Dec 5th 2007, 03:13 AM
mathgirl
okay...thank you!!! :)