Problem For Putnam Class
Point of post: In this post I’ll solve a fairly easy (though it took me a bit to see exactly now to do it) linear algebra problem. This is for a Putnam class I’m taking, where we’re encouraged to do as many problems as one can out of the six or so posted each week. This was one of the “easier” ones (they’re two categories, easier problems and past Putnam problems).
Problem: Let be an -dimensional vector space and be a linear transformation. Prove that if has eigenvectors, each which are linearly independent, then for some .
Proof: Denote the eigenvectors as and their corresponding eigenvalues as . Let and with be arbitrary. By possible relabeling we may assume that and . But, we know that is linearly independent and so (Horn, 46)
But, since is also linearly independent we may appeal to the same idea and see that
from where it follows by the transitivity of the similarity relation that
Thus, since the trace of a matrix is invariant under similarity we may conclude that
and so by cancellation we get that . Since (by relabeling) the eigenvalues were arbitrary we may conclude that . Therefore, we can see that since
we may conclude that
from where the conclusion follows.
1. Horn, Roger A., and Charles R. Johnson. Matrix Analysis. Cambridge [u.a.: Cambridge Univ., 2009. Print.
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