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As we have eigenspace , I have also read that eigenvector can not be a zero vector , so how has it been possible to say eigenspace is there without null vector .

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2 Answers

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The eigenspace contains the zero vector, but the zero vector is often not considered to be an eigenvector (e.g. because if it were, then every linear map would have all elements of the base field as eigenvalues, since $\lambda\cdot0=0$ for all $\lambda$...)

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An eigenvector is nonzero by definition. I like the following reason: Let $T: V \to V$ be linear. Then, we will say that a one-dimensional eigenspace $W \subseteq V$ is a one-dimensional subspace such that $T|_W = \lambda \,\mathrm{id}_W$ for some scalar $\lambda$. This definition makes it clear that a zero-dimensional eigenspace is always uninteresting.

Then, we make a secondary definition of eigenvector, which is an element whose span is a one-dimensional eigenspace.

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