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Math Help - complex conjugates and unitary

  1. #1
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    complex conjugates and unitary

    For M \inM_nxn(C), let the \overline{M} be the matrix such that \overline{(M)}_ij= \overline{M} _ij for all i,j, where \overline{M}_ij is the complex conjugate of M_ij.

    (1) Prove that det \overline{M} = \overline{det(M)}

    (2) A matrix Q in M_nxn(C)is called unitary if QQ*=I, where Q*= \overline{Q^t}. Prove that if Q is a unitary matrix, then |det(Q)|=1
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  2. #2
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    Quote Originally Posted by tn11631 View Post
    For M \inM_nxn(C), let the \overline{M} be the matrix such that \overline{(M)}_ij= \overline{M} _ij for all i,j, where \overline{M}_ij is the complex conjugate of M_ij.

    (1) Prove that det \overline{M} = \overline{det(M)}


    This follows at once from the definition of determinant and because complex conjugation is and additive and multiplicative function


    (2) A matrix Q in M_nxn(C)is called unitary if QQ*=I, where Q*= \overline{Q^t}. Prove that if Q is a unitary matrix, then |det(Q)|=1
    As \det Q^*=\overline{\det Q} ,as was proved in (1), you get \det I=\det (QQ^*)=\det Q\cdot\det Q^* ...etc.

    Tonio
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