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Math Help - Similarity of a matrix

  1. #1
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    Similarity of a matrix

    Let A be an n*n matrix with real entries and A^2+I=0. If n=2k then show that A is similar to the block matrix of the form \left(\begin{array}{cc}0&-I\\I&0\end{array}\right) where I is k*k identity matrix.
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    MHF Contributor Drexel28's Avatar
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    Re: Similarity of a matrix

    Hint: If n=2k then there is a natural ring morphism of \text{Mat}_n(\mathbb{R})\to \text{Mat}_k(\mathbb{C}). Now, the image of your matrix A is a complex matrix with A^2+I=0. This implies that the minimal polynomial of A divides x^2+1, and thus is separable. Thus, A is diagonalizable, and it has eigenvalues \pm i. Now, what?
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  3. #3
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    Re: Similarity of a matrix

    Hi xixi,
    The fact that the matrix is over the reals and not the complex field is critical. The 2 by 2 scalar matrix A with i on the diagonal clearly satisfies A2+I=0 and is certainly not similar to a matrix of the stated form. (Since A commutes with anybody, a conjugate of A is A!)
    Attached is a solution. If you have questions, post your question.

    Similarity of a matrix-mhflinearalgebra8.png
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