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Math Help - Unitary matrices- General form

  1. #1
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    Unitary matrices- General form

    Does anybody know the general form of a 2x2 unitary matrix such that GG*=I. Where I is identity matrix and G* is conjugate traspose of G. I have been told that it is:

    a -b*
    b a*

    Is this correct???
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  2. #2
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    Quote Originally Posted by johnbarkwith View Post
    Does anybody know the general form of a 2x2 unitary matrix such that GG*=I. Where I is identity matrix and G* is conjugate traspose of G. I have been told that it is:

    a -b*
    b a*

    Is this correct???
    That is partially correct. To have that form, the matrix must be in the special unitary group SU(2). This means that in addition to being unitary, the matrix has determinant 1.

    If the matrix \begin{bmatrix}a&c\\b&d\end{bmatrix} is unitary then \begin{bmatrix}a&c\\b&d\end{bmatrix} \begin{bmatrix}\bar{a}&\bar{b}\\\bar{c}&\bar{d}\en  d{bmatrix} = \begin{bmatrix}1&0\\0&1\end{bmatrix}. Compare matrix elements on both sides of that equation, together with the equation ad-bc=1, and you will find that d=\bar{a} and c=-\bar{b}. (I'm using bars rather than stars to denote complex conjugates.)

    The general unitary matrix could have determinant 1, in which case it has the form \begin{bmatrix}a&\bar{b}\\b&-\bar{a}\end{bmatrix}.
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    small correction

    Quote Originally Posted by Opalg View Post
    The general unitary matrix could have determinant 1, in which case it has the form \begin{bmatrix}a&\bar{b}\\b&-\bar{a}\end{bmatrix}.
    Actually.... a general unitary matrix can have determinant any complex number of absolute value 1. So the matrix \begin{bmatrix}a&-\lambda\bar{b}\\b&\lambda\bar{a}\end{bmatrix}
    for |\lambda|=1 is also unitary, with determinant \lambda. (It is also necessary that |a|^2 + |b|^2 = 1.)
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