Sequence of matrices and charecteristic polynomials can you help?

Let M_k be a sequence of complex square matrices, with characteristic polynomials P_k, that converges to a matrix M with characteristic polynomial P. The, does P_k need to converge to P? If so, how can we prove this?

What if, instead of characteristic polynomials, P_k and P are the respective minimal polynomials of the matrices?

Thank you.

2 Answers

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  • 1 decade ago
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    I agree with Apratim's comments concerning the characteristic polynomial. However, the minimal polynomial is not a continuous function of the matrix, so the minimal polynomial of the limit is not necessarily the limit of the minimal polynomials. For example, if M_k = [1, 0; 0, 1+1/k], then P_k = (x-1)(x-1-1/k), which converges to (x-1)^2. But {M_k} converges to the identity matrix, whose minimal polynomial is (x-1).

  • 1 decade ago

    The coefficients of the characteristic polynomial are *continuous* functions of the entries of the matrix. So if a sequence of matrices converges to a limiting matrix, then the sequence of characteristic polynomials also converges, and converges to the characteristic polynomial of the limiting matrix.

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