Polynomials in a matrix |
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Authors: | Morris Newman |
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Institution: | National Bureau of Standards , Washington, D.C., 20234 |
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Abstract: | Let A be a matrixp(x) a polynomial. Put B=p(A). It is shown that necessary and sufficient conditions for A to be a polynomial in B are (i) if λ is any eigenvalue of A, and if some elementary divisor of A corresponding to λ is nonlinear, thenp ′(λ)≠0;and (ii) if λ,μ are distinct eigenvalues of A, then p(λ)p(μ) are also distinct. Here all computations are over some algebraically closed field. |
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