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Math Help - Similar but unitary equivalent

  1. #1
    MHF Contributor
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    Similar but unitary equivalent

    Show that the matrices

    A=\begin{bmatrix}1&3&0\\0&2&4\\0&0&3\end{bmatrix} \ \text{and} \ \begin{bmatrix}1&0&0\\0&2&5\\0&0&3\end{bmatrix}

    are similar and satisfy \sum_{i,j}|a_{i,j}|^2=\sum_{i,j}|b_{i,j}|^2 but are not unitary equivalent.

    I am find it had to find a matrix S such that B=SAS^{-1}.
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  2. #2
    A Plied Mathematician
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    Re: Similar but unitary equivalent

    Suppose you try diagonalizing both matrices? If there is a P s.t. B=PDP^{-1}, and there exists a Q s.t. A=QDQ^{-1}, for the same diagonal matrix D, then P^{-1}BP=D and Q^{-1}AQ=D, so

    P^{-1}BP=Q^{-1}AQ, and hence B=PQ^{-1}AQP^{-1}.

    What is your candidate for S, then?
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