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could any one tell me what is translation? and if a $n\times n$ matrix $A$ commutes with all translation then must it be a circulant matrix? first please tell me the meaning of the question and then explain, please help, it was a question from a past year quals.thank you.

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Does "the question" refer to the second of your two questions? – joriki Feb 3 '13 at 19:09

Here a "translation" means a circulant shift operator of the form $$(x_1,\ldots,x_n)^T \mapsto (x_{k+1},\ldots,x_n,x_1,x_2,\ldots,x_k)^T$$ for some $k$.

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