What is the restriction operator on an eigenspace?

Question

This is the question I am having trouble with:

Let c be a characteristic value of $T$ (a linear operator on $V$) and let $W$ be the space of characteristic vectors associated with the characteristic value of c. What is the restriction operator $T_w$?

My book (Hoffman & Kunze) gives a proof with matrices that $T_w$ divides the mimimum polynomial for $T$. It also states that $T_w\alpha$ = $T\alpha$ for all $\alpha$ in W, but $T_w$ is very different from $T$ in that $T_w$ is an operator on $W$, not $V$ like $T$. So how do I find $T_w$?

Thanks in advance.

Answer 1

Given a linear operator $T$ on vector space $V$, and a linear subspace $W$ of $V$, the restriction $T_W$ of $T$ to $W$ is the linear operator on $W$ defined by $T_W w = T w$ for $w \in W$. So if $Tw = \alpha w$ for $w \in W$, then $T_W w = \alpha w$ as well, i.e. $T_W$ is $\alpha$ times the identity operator.

Answer 2

i will take an example so that i can be specific. let $e_1 = (1,0,0,0)^T, e_2,e_3$ and $e_4$ be the standard basis for $R^4.$ consider the linear operator $T:R^4 \to R^4$ defined by $Te_1 = 0, Te_2 = e_1, Te_3 = e_3$ and $Te_4 = e_3 + e_4,$ subspace $W = span \{e_1, e_2 \}.$

$T$ is represented by $A = \pmatrix{0 & 1 & 0 & 0\cr0 & 0 & 0 & 0 \cr0&0&1&1\cr0&0&0&1}$ with respect to the standard basis. the restriction of $T_w:W \to W$ of $T$ to $W$ is represented by the matrix $A_w = \pmatrix{0&1\cr0&0}$ with respect to the basis $\{e_1, e_2\}.$

the char poly of $A$ is $(\lambda^2(\lambda - 1)^2$ and of $A-w$ is $\lambda^2$