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Math Help - Rank one matrix has a nonzero eigenvalue?

  1. #1
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    Rank one matrix has a nonzero eigenvalue?

    How can I show that a rank one matrix has a nonzero eigenvalue?
    It seems obvious, but how to prove it?
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  2. #2
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    Re: Rank one matrix has a nonzero eigenvalue?

    Not so obvious in the case A:=\begin{pmatrix}0&1\\0&0\end{pmatrix}.
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    Re: Rank one matrix has a nonzero eigenvalue?

    Oops! Then, is it true that not all rank one matrices are diagonalizable?
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    Super Member girdav's Avatar
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    Re: Rank one matrix has a nonzero eigenvalue?

    Yes, the A in my previous post showed.
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    Re: Rank one matrix has a nonzero eigenvalue?

    If it is rank one and symmetric matrix, how do we show it is diagonalizable?
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    Super Member girdav's Avatar
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    Re: Rank one matrix has a nonzero eigenvalue?

    It's not diagonalizable.
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  7. #7
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    Re: Rank one matrix has a nonzero eigenvalue?

    Maybe my expression is too poor.
    I am talking about another matrix now, that is having rank one and it is symmetric, with these two criteria, it should be diagonalizable, and how to prove it?
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  8. #8
    Super Member girdav's Avatar
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    Re: Rank one matrix has a nonzero eigenvalue?

    Sorry, it's not your expression but my reading which is poor. A symmetric matrix with real entries is diagonalizable (it's a general result, known as spectral theorem, and doesn't use the fact that the rank is 1), and the involved diagonal matrix has rank 1. So there is in this case a non-zero eigenvalue.
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    Re: Rank one matrix has a nonzero eigenvalue?

    Quote Originally Posted by girdav View Post
    Not so obvious in the case A:=\begin{pmatrix}0&1\\0&0\end{pmatrix}.
    Both eigenvalues of this matrix are 0.
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    Re: Rank one matrix has a nonzero eigenvalue?

    Yes, that was the point of girdav's example.
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    Re: Rank one matrix has a nonzero eigenvalue?

    Quote Originally Posted by HallsofIvy View Post
    Yes, that was the point of girdav's example.
    Ah!
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