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Thread: Unitary matrices

  1. #1
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    Unitary matrices

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


    2.Let $\displaystyle A\in M_n(C)$ be a matrice with $\displaystyle \lambda_1,\lambda_2...\lambda_n$ eigenvalues so that
    $\displaystyle |\lambda_1|...|\lambda_n|=1$.
    Is $\displaystyle A$ 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 $\displaystyle U$ and Diagonal matrice $\displaystyle D$ That $\displaystyle U^*AU=D$ For $\displaystyle A=\begin{bmatrix}1 & 1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$
    The matrix $\displaystyle A$ is nornal, that is $\displaystyle AA^*=A^*A$ . This mean that eigenvectors associated to distinct eigenvalues are orthogonal. The eigenvalues of $\displaystyle A$ in this case are simple, so find the corresponding eigenvectors $\displaystyle u_1,u_2,u_3$ and divide between the norm to obtain $\displaystyle e_1,e_2,e_3$. Then,

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

    Fernando Revilla
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by cursedbg View Post
    1. Find Unitary matrice $\displaystyle U$ and Diagonal matrice $\displaystyle D$ That $\displaystyle U^*AU=D$ For $\displaystyle A=\begin{bmatrix}1 & 1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & 1\end{bmatrix}$

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

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

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

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

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

    Fernando Revilla
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