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Math Help - Eigenvalue question

  1. #1
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    Eigenvalue question

    Hi.

    Why is it a stochastic matrix always has the eigenvalue 1? I understand that the absolute value of a stochastic matrix can not exceed 1 since the determinant of the matrix has to be 1 or less than 1. But why will one eigenvalue always be 1?
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  2. #2
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    Quote Originally Posted by krje1980 View Post
    Hi.

    Why is it a stochastic matrix always has the eigenvalue 1? I understand that the absolute value of a stochastic matrix can not exceed 1 since the determinant of the matrix has to be 1 or less than 1. But why will one eigenvalue always be 1?
    The vector with each component equal to 1 is an eigenvector for the transpose of any stochastic matrix, with eigenvalue 1. But the eigenvalues of a matrix and its transpose are the same.
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  3. #3
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    Ah, that makes sense. Cheers! Thanks a lot!
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