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Thread: complex conjugates and unitary

  1. #1
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    complex conjugates and unitary

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

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

    (2) A matrix Q in M_nxn(C)is called unitary if QQ*=I, where Q*=$\displaystyle \overline{Q^t}$. 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$\displaystyle \in$M_nxn(C), let the $\displaystyle \overline{M}$ be the matrix such that $\displaystyle \overline{(M)}$_ij=$\displaystyle \overline{M}$ _ij for all i,j, where $\displaystyle \overline{M}$_ij is the complex conjugate of M_ij.

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


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

    Tonio
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