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Math Help - Eigenvalues of Unitary Matrix proof

  1. #1
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    Eigenvalues of Unitary Matrix proof

    Unitary is UU*=I
    U* is transpose conjugate

    Prove that if a matrix U is unitary, then all eigenvalues of U have absolute value of 1.

    Uv= \lambda v

    U^* Uv=\lambda U^*v

    v= \lambda U^* v

    v/\lambda=U^* v
    so v is also a eigenvector for U* with eigenvalue of  1/\lambda.
    Not sure how to continue?
    any help would be appreciated
    Thanks in advance
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  2. #2
    Senior Member
    Joined
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    Quote Originally Posted by firebio View Post
    Unitary is UU*=I
    U* is transpose conjugate

    Prove that if a matrix U is unitary, then all eigenvalues of U have absolute value of 1.

    Uv= \lambda v

    U^* Uv=\lambda U^*v

    v= \lambda U^* v

    v/\lambda=U^* v
    so v is also a eigenvector for U* with eigenvalue of  1/\lambda.
    Not sure how to continue?
    any help would be appreciated
    Thanks in advance
    The eigenvalues of U^* are the complex conjugates of the eigenvalues of U (link).

    You have

    Uv = \lambda v,
    U^*v = (1/\lambda)v.

    Thus, \lambda = 1/\bar{\lambda} \Leftrightarrow \lambda \bar{\lambda} =1 \Leftrightarrow |\lambda|^2=1 .
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