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Thread: Eigenvalues and Eigenvectors

  1. #1
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    Eigenvalues and Eigenvectors

    Find a basis B for the domain of T such that the matix of T relative to B is diagonal.
    given..
    T: P1 -> P1
    T(a + bx) = a + (a + 2b)x

    I got confused with this hw problem. Thank you.
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  2. #2
    Super Member Rebesques's Avatar
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    Identify the polynomial $\displaystyle a+bx$ with the vector $\displaystyle (a \ b)^t$ (transposed, can't bother texing marices ). Then T acts like multiplication by the matrix $\displaystyle T=
    (T_1 \ T_2), \ T_1=(1 \ 1)^t , \ T_2=(0 \ 2)^t $.

    How to diagonalize this? Solve $\displaystyle det(T-\lambda I)=0$ for $\displaystyle \lambda $to get the eigenvalues 1 and 2. The corresponding eigenspaces come from $\displaystyle V(1)=\{a+bx: T(a+bx)=a+bx\}$ and $\displaystyle V(2)=\{a+bx: T(a+bx)=2(a+bx)\}$. Gladly, each space has one unit eigenvector, and they are orthogonal by theory. Arrange these in a matrix P, and you are done.
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