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    • Thread Starter

    Given a matrix A with complex entries, how do I find the invertible matrix P such that P^-1 * A * P is upper triangular? I have seen the proof of the fact that every matrix is similar to an upper triangular matrix, but the proof was by induction and so I don't think that it can be used to construct the matrix.
    • PS Helper

    PS Helper
    The idea is that if you can find an eigenvector then the image of that eigenvector is going to be a multiple of itself, so choose that to be your first basis vector; then you have a column that looks like \begin{matrix} \lambda \\ 0 \\ \vdots \\ 0 \end{matrix}. Then you need to proceed to find linearly independent vectors v_k such that for each k, when you apply your matrix to it what you get is some linear combination of v_1, \dots, v_k (i.e. and not v_{k+1} , \dots, v_n). Then the matrix must be upper-triangular with respect to the resulting basis.
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Updated: April 8, 2011

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