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It is common fashion to try to formulate results about matrices in a basis-free way, using linear algebra. What are some good examples of situations where this is impossible? I illustrate what I have in mind with the following two examples:

  1. The Perron-Frobenius theorem. Here the nonnegativity is essential, and the conclusions are substantial.

  2. Stochastic matrices, in general. Here again, the crucial thing is nonnegativity.

Are there other such situations where going basis-free is an impossible task? The examples I gave utilize nonnegativity of entries; if there are examples utilizing some other fact, that will be particularly nice.

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I don't get exactly what basis-free means. For example, is matrix diagonalization $A = S^{-1}\Lambda S$ basis-free? –  user13838 Sep 19 '11 at 9:04
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