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Math Help - non-diagonalizable matrix help?

  1. #1
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    non-diagonalizable matrix help?

    Hello,

    I've been asked to find a 3x3 matrix A such that

    1) A^3 - 4A^2 + 5A - 2I = 0

    and

    2) The eigenvectors of A do not span R^3

    I believe the second condition implies that A is not diagonalizable, which means it does not contain a complete set of eigenvectors (I am only aware that nilpotent matrices fall under this category, are there others?) and I know that 1) can be expressed as:
    (A - I)^2(A-2I)

    but I cannot figure out how to work this out. Can anyone help?
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  2. #2
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    Quote Originally Posted by drangelpsyche View Post
    Hello,

    I've been asked to find a 3x3 matrix A such that

    1) A^3 - 4A^2 + 5A - 2I = 0

    and

    2) The eigenvectors of A do not span R^3

    I believe the second condition implies that A is not diagonalizable, which means it does not contain a complete set of eigenvectors (I am only aware that nilpotent matrices fall under this category, are there others?) and I know that 1) can be expressed as:
    (A - I)^2(A-2I)

    but I cannot figure out how to work this out. Can anyone help?

    So the characteristic pol. of A is p_A(x)=(x-1)^2(x-2) and its minimal pol. MUST be the same, otherwise, as the char. pol. and the min. pol. have the same irreducible factors, the min. pol would be m_A(x)=(x-1)(x-2) , but then A would be diagonalizable as its min. pol. would be the product of different linear factors...
    Now, check that the following matrix does the trick:

    A=\left(\begin{array}{rrr}0&1&0\\0&0&1\\2&-5&4\end{array}\right)

    This matrix is known as the companion matrix of the pol. f(x) and it's used to prove that for any given monic polynomial there exists a square matrix whose char. pol. is that given monic pol.

    Tonio
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  3. #3
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    Thank you very much for your help!
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