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Math Help - Unitarily equivalent matrices and singular values

  1. #1
    Member Mollier's Avatar
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    Unitarily equivalent matrices and singular values

    Hi.

    problem:

    Two matrices A,B\in\mathbb{C}^{m\times m} are unitarily equivalent if A=QBQ^{*} for some unitary Q\in\mathbb{C}^{m\times m}.
    Is it true or false that A and B are unitarily equivalent if and only if they have the same singular values?

    attempt:

    First direction: If A=QBQ^{*}, do A and B have the same singular values?

    Let,
    A=U_A \Sigma_A V^{*}_A, and let,
    B=U_B \Sigma_B V^{*}_B.

    Since A=QBQ^{*}, we have that;
    QBQ^{*}=U_A \Sigma_A V^{*}_A \Rightarrow B=(Q^{*}U_A)\Sigma_A(V^{*}_AQ).
    This shows that B has the same singular values as A.

    Second direction:
    If A and B have the same singular values, are they unitarily equivalent?

    Let,
    A=U_A \Sigma V^{*}_A, and let,
    B=U_B \Sigma V^{*}_B.

    Then,

    U^{*}_A A V_A = U^{*}_B B V_B and so,

    A=(U_A U^{*}_B)B(V_B V^{*}_A).

    I would like to show that U_BU^{*}_A=V_BV^{*}_A, but do not know how.

    Interesting fact is that AA^{*}=(U_AU^{*}_B)BB^{*}(U_BU^{*}_A). Maybe I could use that in some way?

    All suggestions are appreciated, thanks.
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  2. #2
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    For the "second direction", look at the matrices \begin{bmatrix}1&0\\0&1\end{bmatrix} and \begin{bmatrix}1&1\\0&1\end{bmatrix}.
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  3. #3
    Member Mollier's Avatar
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    Ok, first of all I see that they have the same singular values.
    I'll try this first:

    <br /> <br />
\begin{bmatrix}1&1\\0&1\end{bmatrix}=Q \begin{bmatrix}1&0\\0&1\end{bmatrix}Q^{*}=QQ^{*}=I<br />
    So that does not work.

    \begin{bmatrix}1&0\\0&1\end{bmatrix}=Q  \begin{bmatrix}1&1\\0&1\end{bmatrix}Q^{*}<br />
    does not work either, because then
    say,    \begin{bmatrix}1&1\\0&1\end{bmatrix}Q^{*}<br />
would have to be the inverse of Q, which can not be the case.

    So the answer to the "second direction" is no. Even though a counter example is great, is there a way to come to that conclusion without using an example?
    Perhaps by adding something to the work I have already done?

    Thank you very much.
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  4. #4
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    Quote Originally Posted by Mollier View Post
    Ok, first of all I see that they have the same singular values.
    I'll try this first:

    \begin{bmatrix}1&1\\0&1\end{bmatrix}=Q \begin{bmatrix}1&0\\0&1\end{bmatrix}Q^{*}=QQ^{*}=I<br />
    So that does not work.

    \begin{bmatrix}1&0\\0&1\end{bmatrix}=Q  \begin{bmatrix}1&1\\0&1\end{bmatrix}Q^{*}<br />
    does not work either, because then
    say,    \begin{bmatrix}1&1\\0&1\end{bmatrix}Q^{*}<br />
would have to be the inverse of Q, which can not be the case.

    So the answer to the "second direction" is no. Even though a counter example is great, is there a way to come to that conclusion without using an example?
    Perhaps by adding something to the work I have already done?
    The only thing that stops two matrices with the same eigenvalues (= singular values) from being unitarily equivalent is if their eigenspaces have different dimensions. What that means in plain English is that if for example a matrix has a repeated eigenvalue, then there may be only one linearly independent eigenvector for that eigenvalue, or there may be more than one. Two such matrices are unitarily equivalent if and only if each repeated eigenvalue has the same number of linearly independent eigenvectors in both matrices. The dimensions of the eigenspaces are invariant under unitary equivalence, so unitarily equivalent matrices must have eigenspaces of the same dimension.

    In the example that I suggested, the 22 identity matrix I has a repeated eigenvalue 1. It has a two-dimensional eigenspace, because every vector x in the space satisfies Ix=x and is therefore an eigenvector for the eigenvalue 1. The matrix A = \begin{bmatrix}1&1\\0&1\end{bmatrix} also has a repeated eigenvalue 1, but the only linearly independent eigenvector is \begin{bmatrix}1\\0\end{bmatrix}. So the eigenspace is one-dimensional, and that stops A from being unitarily equivalent to I.
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  5. #5
    Member Mollier's Avatar
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    Ah, great stuff.

    So if A and B are normal matrices and have the same eigenvalues, then they are unitarily equivalent.
    If they are normal, then by the spectral theorem;

    A=P\Lambda P^{*} and B=Q\Lambda Q^{*}.

    Then,

    B=QP^{*}APQ^{*}.

    Thanks!
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