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Math Help - Invertible matrix

  1. #1
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    Invertible matrix

    I'm supposed to find the matrix P here. I've found the characteristic polynomial p(lamda) = -x - x^2 + x^3 + x^4 (just writing x instead of lamda). And I found the roots:

    x=0
    x=1
    x=-1

    But I can't make sense of what the book does next to find P... Any help would be very appreciated! =)
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  2. #2
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    They don't tell you any more information? When you say "find the matrix P", do you mean find the 4x4 matrix with all of its values? Because I think that is impossible just given the characteristic polynomial. Multiple 4x4 matrices can yield that same polynomial. Also there are two -1 roots, just fyi...
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    Quote Originally Posted by gralla55 View Post
    I'm supposed to find the matrix P here. I've found the characteristic polynomial p(lamda) = -x - x^2 + x^3 + x^4 (just writing x instead of lamda). And I found the roots:

    x=0
    x=1
    x=-1

    But I can't make sense of what the book does next to find P... Any help would be very appreciated! =)
    Are you trying to diagonalize a given matrix? If so you must find n linearly independent eigenvectors of that nxn matrix, say p_1,p_2,...,p_n and form the matrix P=[p_1, p_2, ... p_n]. The matrix P^{-1}AP will be diagonal and will have the eigenvalues corresponding to p_1,p_2,...,p_n, respectively, as its successive diagonal entries.
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  4. #4
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    This is a fourth degree polynomial. There needs to be four solutions. What was the matrix that gave you those lambda?

    \begin{bmatrix}<br />
a_{11}-\lambda & a_{12} & a_{13} & a_{14}\\ <br />
a_{21} & a_{22}-\lambda & a_{23} & a_{24}\\ <br />
a_{31} & a_{32} & a_{33}-\lambda & a_{34}\\ <br />
a_{41} & a_{42} & a_{43} & a_{44}-\lambda<br />
\end{bmatrix}

    To obtain your eigenvectors, you need to plug in each lambda and then rref the matrix. Then do that for the other eigenvalues until you have 4 eigenvectors. If you don't obtain 4 eigenvectors, this matrix isn't diagonalizable.
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  5. #5
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    Please write the entire problem! I suspect the respondents are correct, that you are given a matrix, A, and are asked to find the matrix P such that A= PDP^{-1} where D is a diagonal matrix having the eignvalues of A on its diagonal. But what P is depends strongly on what A is- and different matrices can have the same characteristic polynomial.

    dwsmith- he did give all roots. -1 is a double root.
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  6. #6
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    Quote Originally Posted by HallsofIvy View Post
    Please write the entire problem! I suspect the respondents are correct, that you are given a matrix, A, and are asked to find the matrix P such that A= PDP^{-1} where D is a diagonal matrix having the eignvalues of A on its diagonal. But what P is depends strongly on what A is- and different matrices can have the same characteristic polynomial.

    dwsmith- he did give all roots. -1 is a double root.
    I understand but it should have been noted.
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