I am studying linear algebra and got confused in defining eigen space corresponding to eigen value.The thing wondering me is the same thing defined in two different books in different manners.let $\lambda$ be eigen value of matrix $A$ of order n over the field $\mathbb{F}$ then Hoffman & kunze defines eigen space as follows:
Definition:The collection of all $x\in\mathbb{F^n}$ such that $Ax=\lambda x$ is called the eigenspace associated with $\lambda$.
While in another book A first course in module theory by M.E Keating,he defines eigenspace as follows:
Definition:Given an n x n matrix $A$ over a field $\mathbb{F}$, an eigenspace for $A$ is a nonzero
subspace $U$ of $\mathbb{F^n}$ with the property that $$Ax=\lambda x$$ $\forall x\in U$
I think the two definitions are not equivalent:
Let $x_1,x_2$ be two linearly independent eigenvectors corresponding to eigen value $\lambda$,then according to first definition a subspace generated by $x_1,x_2$ is the eigen space of $A$ corresponding to eigen value $\lambda$ while according to definition-2 there are three eigen space each of which generated by
(1)$x_1$ only,
(2)$x_2$ only,
(3) $x_1$ and $x_2$
Am I right in my understanding?since my graduation days I am familiar with first definition and used it in my problems,could anybody help me in understanding
Thanks in advance
Best Answer
You are right: they are not equivalent. The first definition is the usual one. The second one is more general: if $F_\lambda$ is the eigenspace corresponding to $\lambda$ (with respect to the first definition), then the author of the second definition is saying that any non-zero subspace of $F_\lambda$ is an eigenspace for $A$ corresponding to the eigenvalue $\lambda$.