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Why are trivial solutions "wrong"?

For example, if I'm solving a PDE and the eigenvalue being zero implies that the solution to the PDE is identically $0$, why do we say that the eigenvalue cannot equal $0$?

Jimm
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  • When did you hear that an eigenvalue cannot equal $0$? – 5xum Dec 11 '15 at 19:22
  • They're not wrong, just not very interesting. If you assume that the eigenvalue ≠ 0, then you can use that assumption to find other solutions than the trivial ones. – Alice Ryhl Dec 11 '15 at 19:23
  • The solution to a PDE (or any linear system) that is zero is not an eigensolution (or eigenvector). Note that $0 = \lambda 0$ is true for any $\lambda$, so the zero solution does not help us to characterize what $\lambda$ can be. – hardmath Dec 11 '15 at 19:27
  • 5xum that's not what I said. I am referring specifically to PDE's, specifically to the heat equation with Neumann boundary conditions. – Jimm Dec 11 '15 at 19:32
  • That's kind of what I mean. It still solves it, even if it is "useless." – Jimm Dec 11 '15 at 19:33
  • For a linear system, every value would be an eigenvalue if you did not require non-zero solutions. Every value would be an eigenvalue of a matrix, for example. Not a very interesting concept. – Disintegrating By Parts Dec 12 '15 at 16:59

1 Answers1

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The definition of eigenvalue requires that the associated eigenvector (or eigenfunction) be non-zero.

Barry Cipra
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