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Math Help - Linear Algebra- Normal and Self-Adjoint Operators

  1. #1
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    Exclamation 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!!!
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  2. #2
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    You can express T by means of a matrix: 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 \begin{vmatrix}2-\lambda&-2\\-2&5-\lambda\end{vmatrix}. Can you take it from there?
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    Quote Originally Posted by Opalg View Post
    You can express T by means of a matrix: 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 \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!
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    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
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  5. #5
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    Quote Originally Posted by mathgirl View Post
    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.
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    okay...thank you!!!
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