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  1. #1
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    Linear Algebra help

    Suppose that A is a square matrix and it has two distinct eigenvalues, lamda 1 and lamda 2. and that dim(E_lamda1)=n-1. Prove that A is diagonalizable.
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  2. #2
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    Suppose that A is a square matrix and it has two distinct eigenvalues, lamda 1 and lamda 2. and that dim(E_lamda1)=n-1. Prove that A is diagonalizable.

    GIVEN INFORMATION:

    A is a square matrix ( n x n ).
    A has two distinct eigenvalues,  \lambda_{1} and  \lambda_{2} .
     dim(E_{\lambda_{1}}) = n-1

    Proof:

    For an n x n matrix, the dimensions of the eigenspaces must add up to n for the matrix to be diagonalizable. We are given  dim(E_{\lambda_{1}}) = n-1 . Furthermore, we know that  dim(E_{\lambda_{2}}) \geq 1 and  dim(E_{\lambda_{1}}) + dim(E_{\lambda_{1}}) \leq n . Thus,  dim(E_{\lambda_{2}}) = 1 .

    So, finally,  dim(E_{\lambda_{1}}) + dim(E_{\lambda_{2}}) = (n-1) + 1 = n . We have proven that A is diagonalizable.

    -Andy
    Last edited by abender; July 31st 2008 at 08:41 PM.
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