For an operator $T \in \mathscr{L}(\mathbf{V})$, a $T$-invariant subspace is subspace $\mathbf{U} \leq \mathbf{V}$ whose image under $T$ is a subspace of itself.
$$ T\restriction_{\mathbf{U}} \in \mathscr{L}(\mathbf{U}) \hspace{15pt} \equiv \hspace{15pt} \forall u \in U, Tu \in U $$
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Lemma. The null space and range of $T \in \mathscr{L}(\mathbf{V})$ are $T$-invariant.
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Lemma. For $T \in \mathscr{L}(\mathbf{V})$ and $T$-invariant subspaces $\mathbf{U}, \mathbf{W} \leq \mathbf{V}$ s.t. $\mathbf{U} \oplus \mathbf{W} = \mathbf{V}$, with bases $\beta$ and $\gamma$ respectively, the following is true:
$$ [T]_{\beta +\gamma} = \begin{pmatrix} [T\restriction {\mathbf{U}}]{\beta} & 0 \\ 0 & [T \restriction {\mathbf{W}}]\gamma\end{pmatrix} $$
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Theorem. For any polynomial $p \in P(F)$ and operator $T \in \mathscr{L}(\mathbf{V})$, the null space and range of $p(T)$ are $T$-invariant.
A scalar $\lambda \in F$ is an eigenvalue of $T \in \mathscr{L}(\mathbf{V})$ if there exists a nonzero vector $v \in V$ such that,
$$ Tv=\lambda v $$
Moreover, the set of eigenvalues of $T$ is called its spectrum $\sigma (T)$ and the vector space of eigenvectors $v$ associated with an eigenvalue $\lambda$ is the eigenspace associated with $\lambda$, denoted $E(\lambda, T)$.
$$ E(\lambda , T) := \{v \in V : Tv = \lambda v \} = \text{null }(T-\lambda I) $$
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Lemma. The number of eigenvalues of $T \in \mathscr{L}(\mathbf{V})$ is at most the dimension of $\mathbf{V}$.
$$ |\sigma (T)| \leq \dim \mathbf{V} $$
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Lemma. If $T \in \mathscr{L}(\mathbf{V})$ has distinct eigenvalues $\lambda_1 , \dots, \lambda_m$, then the sum of eigenspaces is direct and is bounded in dimension by the dimension of $\mathbf{V}$.
$$ \sum _{i=1}^m E(\lambda_i,T) = \bigoplus _{i=1}^m E(\lambda_i, T) \text{ with } \dim \bigoplus _{i=1}^m E(\lambda_i, T) \leq \dim \mathbf{V} $$
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Theorem. Any list of eigenvectors of $T\in \mathscr{L}(\mathbf{V})$, each corresponding to distinct eigenvalues, is linearly independent.
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Theorem. For a nonzero, finite-dimensional vector space $\mathbf{V}$, algebraically closed field $F$, and $T \in \mathscr{L} (\mathbf{V})$, the spectrum of $T$ is nonempty.
$$ \forall T \in \mathscr{L}(\mathbf{V}) , \space \sigma(T) \neq \empty $$