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An n by n matrix A is periodic if Am = I for some m>0 (i.e., a positive power of A is equal to the identity matrix). The period of A is the smallest positive number m for which Am = I.
Let A and B be n by n periodic matrices with coefficients in the real numbers, R. We say that A and B are topologically similar if there is a homeomorphism (i.e., a continuous, one-to-one and onto function with a continuous inverse) h:Rn -> Rn from Euclidean n-space Rn to itself, such that h(Ax)=Bh(x) for all x in Rn.
If the homeomorphism h were a linear transformation, it would be induced by an invertible matrix, C, and the equation h(Ax)=Bh(x) then shows that CA=BC, so that A and B are linearly similar.
For periodic matrices, linear similarity is easy to detect. For instance, the theory of Jordan canonical form shows that periodic matrices must be diagonalizable over the complex numbers. (Jordan blocks of size bigger than one are not periodic.) And the theory of rational canonical form shows that real matrices are similar over the reals if and only if they are similar over the complex numbers. Thus, A and B are linearly similar if and only if they have the same complex eigenvalues in the same multiplicities (i.e., if their characteristic polynomials are equal).
A deeper argument shows that periodic matrices are linearly similar if and only if they have the same trace.
De Rham, in 1935, conjectured that topologically similar periodic matrices must be linearly similar.
De Rham's conjecture is false. The first counterexamples were found by Cappell and Shaneson in 1979.
This sets up the nonlinear similarity problem: We say that A and B are nonlinearly similar if they are topologically similar but not linearly similar. The Non-linear Similarity Problem is to give precise conditions on the characteristic polynomials of A and B that determine whether or not A and B are non-linearly similar.
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