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Math Help - Unitary matrices

  1. #1
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    Unitary matrices

    1. Find Unitary matrice U and Diagonal matrice D That U^*AU=D
    For A=\begin{bmatrix}1 & 1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}


    2.Let A\in M_n(C) be a matrice with \lambda_1,\lambda_2...\lambda_n eigenvalues so that
    |\lambda_1|...|\lambda_n|=1.
    Is A Unitary?

    So I've got no idea how to approach these problems.
    Thanks in advance.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by cursedbg View Post
    1. Find Unitary matrice U and Diagonal matrice D That U^*AU=D For A=\begin{bmatrix}1 & 1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}
    The matrix A is nornal, that is AA^*=A^*A . This mean that eigenvectors associated to distinct eigenvalues are orthogonal. The eigenvalues of A in this case are simple, so find the corresponding eigenvectors u_1,u_2,u_3 and divide between the norm to obtain e_1,e_2,e_3. Then,

    U=[e_1,\;e_2,\;e_3]

    Fernando Revilla
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    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by cursedbg View Post
    1. Find Unitary matrice U and Diagonal matrice D That U^*AU=D For A=\begin{bmatrix}1 & 1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}

    The matrix A is nornal, that is AA^*=A^*A . This mean that eigenvectors associated to distinct eigenvalues are orthogonal. The eigenvalues of A in this case are simple, so find the corresponding eigenvectors u_1,u_2,u_3 and divide between the norm to obtain e_1,e_2,e_3. Then,

    U=[e_1,\;e_2,\;e_3]

    Fernando Revilla

    Edited: Sorry I published by mistake twice the same post.
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by cursedbg View Post
    2.Let A\in M_n(C) be a matrice with \lambda_1,\lambda_2...\lambda_n eigenvalues so that
    |\lambda_1|...|\lambda_n|=1. Is A Unitary?

    No, it isn't. Choose as a counterexample:

    A=\begin{bmatrix}{1}&{1}\\{0}&{1}\end{bmatrix}

    Fernando Revilla
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