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Math Help - set of all eigen value of square matrix

  1. #1
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    Cool set of all eigen value of square matrix

    Given any square matrix A. Let \sigma(A) be the set of all eigen value of A, then \sigma\left(\left[\begin{matrix} A & A \\ A & A \end{matrix} \right] \right) = ...
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    Quote Originally Posted by GTK X Hunter View Post
    Given any square matrix A. Let \sigma(A) be the set of all eigen value of A, then \sigma\left(\left[\begin{matrix} A & A \\ A & A \end{matrix} \right] \right) = ...
    Hint: If Ax = \lambda x then what are \begin{bmatrix} A & A \\ A & A \end{bmatrix}\begin{bmatrix} x \\ x \end{bmatrix} and \begin{bmatrix} A & A \\ A & A \end{bmatrix}\begin{bmatrix} x \\ -x \end{bmatrix} ?
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    what i do is assuming A = \left[\begin{matrix}1&2\\5&4\end{matrix}\right] \ni \lambda_1=-1,\lambda_2=6 then form a matrix B=\left[\begin{matrix} A & A \\ A & A \end{matrix} \right] then try to find the eigen values. But it's not easy >_<
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    Quote Originally Posted by GTK X Hunter View Post
    what i do is assuming A = \left[\begin{matrix}1&2\\5&4\end{matrix}\right] \ni \lambda_1=-1,\lambda_2=6 then form a matrix B=\left[\begin{matrix} A & A \\ A & A \end{matrix} \right] then try to find the eigen values. But it's not easy >_<
    The eigenvectors of A are \begin{bmatrix}1\\-1\end{bmatrix} (for the eigenvalue 1) and \begin{bmatrix}2\\5\end{bmatrix} (for the eigenvalue 6). According to my previous comment, the eigenvectors for B are \begin{bmatrix}1\\-1\\1\\-1\end{bmatrix},\ \begin{bmatrix}1\\-1\\-1\\1\end{bmatrix},\ \begin{bmatrix}2\\5\\2\\5\end{bmatrix},\ \begin{bmatrix}2\\5\\-2\\-5\end{bmatrix}. From that, you can check that the eigenvalues are 0 (double eigenvalue), 1 and 6.
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    Quote Originally Posted by GTK X Hunter View Post
    what i do is assuming A = \left[\begin{matrix}1&2\\5&4\end{matrix}\right] \ni \lambda_1=-1,\lambda_2=6 then form a matrix B=\left[\begin{matrix} A & A \\ A & A \end{matrix} \right] then try to find the eigen values. But it's not easy >_<
    You do understand, I hope, that showing this for this one matrix does NOT show it is true for any square matrix and so does not answer the question?

    Look closely at Opalg's very good suggestion.
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